Jazz is jumping. It's in nightclubs and record stores, on TV commercials and at the movies, and even on PBS, where Ken Burns's marathon documentary series, for all its gaping holes (no, Virginia, jazz did not come to a screeching halt in 1959), nevertheless succeeded in introducing a new generation of listeners to the soul-satisfying sounds of Louis Armstrong, Charlie Parker and countless other musical titans.

Maybe you caught an episode or two of "Jazz," or heard Diana Krall singing "The Look of Love" on the radio last winter, and started to wonder what you'd been missing. And maybe you told a jazz buff about your belated discovery and received a few snooty remarks in return. Alas, too many jazz buffs are just like that. They behave as if America's only indigenous art form were a private club with guards posted at the door to chase off the curious.

Don't let the snobs scare you. The secret of jazz is that there is no secret. It's for everyone, from first-timers to connoisseurs. All you have to do is listen -- but the trick is to know where to start. I've pushed dozens of friends into the pool, and I'd be happy to give you a shove, too.

So here's a list of 30 must-have jazz CDs, all available in well-stocked shops or online at such Web-based stores as Amazon. Taken together, they cover most of the bases up through the '70s and even beyond. I haven't tried to be fussily encyclopedic, and I've tucked some purely personal favorites in among the recognized classics. Still, everyone on this list is a big-leaguer, and most of the indisputably major players of the past are here, though some, like Earl Hines, Jack Teagarden and Buddy Rich, make only token appearances on albums devoted to other artists.

Don't expect to lose your head over each and every CD I've recommended. Some will hit the target, others the wall. But I promise that if you work through this list from start to finish, you'll come away with a crystal-clear sense of what jazz sounds like -- and not just one particular kind, either, but the whole dazzling array of styles that made up jazz in the 20th century, from New Orleans gumbo all the way to jazz-rock fusion.

That said, I'll lead with my ace in the hole:

Louis Armstrong, "Portrait of the Artist as a Young Man, 1923-1934" (Columbia/Legacy, four CDs). This is the cornerstone, a near-flawless collection of recordings by the single most significant musician ever to play jazz. Armstrong didn't invent jazz, but he perfected it, and his blazing trumpet and gravel-voiced singing changed the way everybody else played and sang it forever after. Go straight to "West End Blues," recorded in 1928, and listen to the proclamatory opening cadenza and efflorescent climactic solo. That says it all, and then some. (If a four-disc set takes too big a bite out of your budget, try "This Is Jazz: Louis Armstrong," which contains 16 of the best sides from "Portrait of the Artist as a Young Man," including "West End Blues.")

Bix Beiderbecke, "At the Jazz Band Ball" (ASV Living Era). Armstrong and Beiderbecke founded the two enduring bloodlines of jazz, hot and cool. Here's the cool. A subtle, elegant melodist with a chimelike cornet tone, Beiderbecke took the pastel harmonies of Debussy and Ravel and made them swing. His classic 1927 version of "Singin' the Blues" was the very first jazz ballad recording and remains one of the very best.

Sidney Bechet, "Really the Blues" (ASV Living Era). Like Armstrong, Bechet hailed from New Orleans, the delivery room of jazz, but he was a little bit older and came into his own a little bit sooner. His pungent style on clarinet and soprano saxophone -- usually bluesy and near-operatic in its frank emotionalism -- is heard to fine effect on this greatest-hits collection.

Jelly Roll Morton, "Birth of the Hot: The Classic Chicago 'Red Hot Peppers' Sessions" (Bluebird). A former whorehouse pianist with the tidy mind of a classical composer, Morton put New Orleans jazz down on paper, turning out quirky, deftly scored small-group miniatures like "Grandpa's Spells" and "The Pearls." All of his key recordings from the late '20s are on this indispensable CD.

"Beyond Category: The Musical Genius of Duke Ellington" (Buddha/BMG, two CDs). What Morton started, Ellington finished. The gorgeously colored three-minute cameos he turned out for his matchless big band were the first jazz compositions to catch the ear of classical critics, and they're as fresh today as they were in the Roaring Twenties and beyond. This two-disc sampler contains 37 well-chosen examples of the Ellington band's work, including such still-familiar standards as "Mood Indigo" and "Take the A Train."

"The Very Best of Fats Waller" (RCA). Out of ragtime came Harlem stride, a jumping brand of piano jazz whose principal exponent, Fats Waller, became even better known in the '30s for his sardonically comic vocals (not to mention the songs he wrote with lyricist Andy Razaf, among them the immortal "Ain't Misbehavin' "). Should your spirits be low, I unhesitatingly guarantee that "The Very Best of Fats Waller" will lift them.

"Ken Burns Jazz: The Definitive Coleman Hawkins" (Polygram). Before Hawkins, the saxophone was strictly from vaudeville. He turned it into the quintessential jazz instrument, playing ballads like "Body and Soul" with a surging romanticism that made him the man to beat on tenor sax throughout the '30s. This CD, one of the many single-disc anthologies intended to supplement the soundtrack of Ken Burns's "Jazz," is an excellent introduction to Hawkins's powerful style.

Benny Goodman, "On the Air (1937-1938)" (Columbia/Legacy, two CDs). The King of Swing was the Depression-era equivalent of a modern rock star, a can-you-top-this clarinet titan who led a ferocious band full of daredevil soloists like trumpeter Harry James and drummer Gene Krupa -- as well as a series of combos featuring pianist Teddy Wilson and vibraphonist Lionel Hampton that were the first racially integrated jazz groups to play in public. These live radio broadcasts from the late '30s, complete with wildly cheering crowds, rank high among the most potent performances from the Swing Era ever to make it onto CD.

Count Basie, "The Best of Early Basie" (GRP). Basie's Kansas City swing was more low-keyed than Goodman's, but no less hot. In addition to his own twinkingly laconic piano playing (he sounded like a minimalist version of Fats Waller, with whom he studied as a teenager), the late-'30s Basie band featured Lester Young, whose sly, understated solos taught a legion of admiring imitators that you didn't have to play tenor sax like Hawkins to be hip.

"Lady Day: The Best of Billie Holiday" (Columbia, two CDs). Holiday's briny-sounding voice may have been less than golden, but what she did with it made her, after Armstrong, the ultimate jazz singer. In dozens of small-group recordings from the '30s -- many made with her close friend Lester Young, and most of the best of them included on this two-CD set -- she took such humdrum tunes as "I Must Have That Man" and ennobled them with a poignant mixture of hopefulness and rue.

Pee Wee Russell, "Take Me to the Land of Jazz" (ASV Living Era). Unlike the infallible Benny Goodman, Russell played clarinet with a gnarly, rasping tone that made you wonder whether he'd ever had a lesson -- but nobody gave a damn, because his surrealistic solos were scaldingly hot. He's identified with "Chicago jazz," the hard-charging style of the '30s and '40s that was later watered down into Dixieland. On this CD, you get the real thing, not its synthetic substitute: "That's A-Plenty," recorded in 1943 with Wild Bill Davison on cornet and George Wettling on drums, sounds like an ammunition dump on fire.

Art Tatum, "The Complete Capitol Recordings" (Capitol, two CDs). Forty-six years after Tatum's untimely death, his name remains a byword for limitless virtuosity -- not just among pianists, either, but wherever jazz musicians of any kind gather. No less a classical keyboard giant than Vladimir Horowitz used to hang out on New York's 52nd Street to hear him do loop-the-loops. Tatum was in his absolute prime when these staggering recordings were made in 1949. Play "Willow Weep for Me" or "Aunt Hagar's Blues" for some unsuspecting pianist and watch him break out in a cold sweat.

"The Best of the Nat King Cole Trio: Instrumental Classics" (Capitol). No, Nat King Cole does not sing on this CD. Years before he metamorphosed into the suavest pop balladeer of the '50s, he was known to jazz fans as the pianist of the King Cole Trio, a superlative combo whose leader was -- and probably still is -- the most underrated instrumentalist of the '40s, fully worthy of comparison with any jazz pianist who ever lived. Listen to him saunter through "What Is This Thing Called Love?" or "Easy Listening Blues" and you'll wish he'd gotten a permanent case of laryngitis.

Woody Herman, "Blowin' Up a Storm: The Columbia Years, 1945-47" (Columbia/Legacy, two CDs). Herman played pretty good clarinet and alto sax, sang the blues like he meant it, and led a big band full of youthful speed demons who galloped out of swing and into bop. The big-band era was in terminal decline when Herman's Herd first hit the road, but you can't tell it from these electrifying performances.

"Ken Burns Jazz: The Definitive Charlie Parker" (Polygram). Bebop starts here. Charlie "Bird" Parker, like Louis Armstrong, changed the sound of jazz, jacking up its rhythmic and harmonic complexity without ever losing sight of the basics. After he came along, you either played alto saxophone like Bird, tried to, or had to explain why you didn't want to. His blues ("Parker's Mood") were down and dirty, his ballads ("Embraceable You") soaringly lyrical, his up-tempo showpieces ("KoKo") awesomely concentrated. For once, the title of this CD is no brag, just fact: If I had to choose Parker's 16 best records, these are the ones I'd pick.

Dizzy Gillespie, "Bebop Professor" (Bluebird). Parker gets most of the ink, but the whip-smart Gillespie deserves equal credit for inventing bebop, and unlike his undisciplined friend and colleague, he was not only a great soloist (on trumpet, which he played with alarmingly bulging cheeks) but also a great bandleader. On "Bebop Professor," you can hear him both in small-group settings and at the helm of the turbulent big band he led in the late '40s and early '50s. Check out the rocking "Manteca," an early example of Afro-Cuban jazz.

"The Best of Thelonious Monk: The Blue Note Years" (Blue Note). Monk played piano with the boppers, but his angular, strangely tilted solos came directly from some other musical solar system. Largely unappreciated in the '40s, he's now considered the most forward-looking jazz composer of his time, and this CD contains the original versions of his best-known pieces.

Miles Davis, "The Complete Birth of the Cool" (Capitol). No other player nudged jazz in as many different directions as Davis. In these recordings, made in 1949 and 1950, the trumpeter led a nine-piece all-star group that collectively lowered the temperature of bebop, opting for concisely lyrical ensembles instead of flag-waving solos. The band included such soon-to-be-celebrated musicians as Gil Evans, Lee Konitz, John Lewis, J.J. Johnson and Gerry Mulligan, but Davis ran the show, set the tone and gets the credit.

"Paul Desmond and the Modern Jazz Quartet" (Sony). John Lewis, who played piano like a cross between Count Basie and Nat Cole, went on to lead the Modern Jazz Quartet, a much-admired combo with a crisp, clean sound and more than a touch of classical rigor. Alto saxophonist Paul Desmond, a wryly witty gent who claimed he wanted to sound like a dry martini, became a star when he paired up with Dave Brubeck to front the most popular jazz group of the '50s. Desmond and the MJQ were mutual admirers, and when they joined forces for a rare 1971 appearance at New York's Town Hall, someone had the sense to turn on a tape recorder and preserve an album's worth of sublimely cool jazz.

"Horace Silver and the Jazz Messengers" (Blue Note). Hard bop was the mirror image of cool jazz -- blunt, aggressive, soulful -- and this is the group, and album, that put it on the map. Silver, the pianist and composer-in-chief, later went off on his own to lead a long string of disciplined yet dynamic quintets; Art Blakey, the take-no-prisoners drummer, took over the Jazz Messengers, fervently preaching the hard-bop gospel until his death in 1990. Still, there's something special about their early collaboration, and Silver's "The Preacher," first recorded in 1954 on "Horace Silver and the Jazz Messengers," hasn't lost any of its funky charm.

Sonny Rollins, "Saxophone Colossus" (Prestige). Don't boggle at the title -- this really is the tenor sax album of the '50s. Rollins's tumbling lines and muscular, iron-toned sound continue to this day to thrill all who hear him in person, but rarely would he again record with so evenly matched a band as Tommy Flanagan on piano, Doug Watkins on bass and Max Roach on drums. Every track is classic, and "Blue Seven" is something more than that, a masterpiece of spontaneous musical organization whose myriad subtleties grow more impressive with multiple listenings.

Charles Mingus, "Mingus Ah Um" (Columbia). When not punching out recalcitrant sidemen (no kidding) or getting himself into every other kind of hot water imaginable, Mingus was a fine bass player and an even better composer, bringing an Ellingtonian richness of color and texture to his small groups. Several of his best-known pieces, including "Goodbye Pork Pie Hat" (a wistful elegy for Lester Young) and "Fables of Faubus" (a snarling anti-tribute to the segregationist governor of Arkansas), are included on this 1959 album, perhaps his most fully realized achievement.

Miles Davis, "Kind of Blue" (Columbia). The biggest-selling jazz album ever, loved by people who know nothing else about jazz save that Davis played it. For musicians, it's a landmark, the record that introduced "modal improvisation," a technique of improvising on scales rather than chordal patterns that was embraced by virtually all post-bop players. Davis solos with a still-startling combination of toughness and fragility, and the rest of the band from 1959 -- Cannonball Adderley, Paul Chambers, Jimmy Cobb, John Coltrane, Bill Evans and Wynton Kelly -- makes more than its share of magic, too.

"The Bill Evans Trio at the Village Vanguard" (Riverside). After "Kind of Blue," Evans quickly established himself as the foremost jazz pianist of the '60s, an introverted poet capable of playing with fiery brilliance whenever he cared to. These performances, taped in 1961 at the best jazz club in New York (it still is), capture Evans at the peak of his powers, buoyed up by the miraculously intuitive support of bassist Scott LaFaro and drummer Paul Motian. I've listened to "My Foolish Heart" at least a hundred times, and it still makes me cry.

Stan Getz, "Getz/Gilberto" (Verve). Getz started out as a Lester Young clone but later developed an immediately identifiable style of his own on tenor saxophone, as satisfying on slower-than-slow ballads as on hell-for-leather swingers. His gentle side was never more in evidence than on this much-loved 1963 album, which teams him with Antonio Carlos Jobim and Joao Gilberto, the Brazilians who brought you the bossa nova. "Getz/Gilberto" introduced "The Girl From Ipanema" and "Quiet Nights of Quiet Stars" to American listeners, and it's as gorgeous as ever 40 years later.

John Coltrane, "A Love Supreme" (Impulse). In the '60s, Coltrane was the tenor saxophonist every other tenor saxophonist wanted to sound like, a volcanic virtuoso who gushed forth long, passionate solos of an almost terrifying intensity. But he also had a spiritual side, and this hugely successful 1964 album, a large-scale composition written by Coltrane for his quartet, recalls "Kind of Blue" in its unexpected combination of uncompromising seriousness and genuine popular appeal.

Wayne Shorter, "Speak No Evil" (Blue Note). Despite the success of "A Love Supreme," Coltrane was and is remembered chiefly as an improviser, whereas tenor saxophonist Shorter, universally acknowledged as a first-class soloist, made his deepest impression with his oblique, strikingly fresh jazz compositions, many of which were recorded by the Miles Davis Quintet during Shorter's tenure with that vastly influential group. He also made solo albums, and "Speak No Evil" features two other Davis sidemen -- Herbie Hancock on piano and Ron Carter on bass -- plus Freddie Hubbard on trumpet and Elvin Jones, John Coltrane's churning drummer. This is the essence of '60s jazz, increasingly abstract yet immediately expressive.

"Gary Burton & Keith Jarrett" (Rhino). As the '60s drew to a close, many younger jazz musicians took an interest in rock, blending its distinctive beat into their ensemble playing to create a new style called, logically enough, fusion. Vibraphonist Burton, a virtuoso of Tatumesque resplendence, was one of the first and best fusionists, and in 1969 he teamed up with pianist Keith Jarrett, who later became world-famous for his unaccompanied solo concerts, for a program of originals that combines jazz, rock and gospel in a thoroughly engaging way.

"This Is Jazz: Weather Report" (Columbia). In 1968, Miles Davis plugged in his trumpet, recording a string of fusion-flavored albums that made his sidemen, among them Keith Jarrett, Chick Corea, John McLaughlin, Tony Williams and Joe Zawinul, stars in their own right. Zawinul and Wayne Shorter then launched their own group, Weather Report, in which synthesizers were used to create a near-orchestral sound superimposed on a driving rock beat. This sampler contains the band's most popular performance, Zawinul's high-stepping "Birdland."

Pat Metheny, "Bright Size Life" (ECM). No jazz guitarist of the '70s has been more widely imitated than Metheny, a resourceful soloist and imaginative composer with an appetite for every possible kind of music. After a tour of duty with Gary Burton, he recorded "Bright Size Life," a 1975 trio album in which jazz and rock are interwoven so completely and compellingly that to call the result fusion is to miss the point. It is, in fact, a masterpiece by a player whose music is so accessible that it tends -- wrongly -- to be underrated.

Needless to say, jazz didn't die out in 1975 any more than it did in 1959. Many of the musicians heard on these 30 albums are alive and well and cooking, and many others have come along since then to carry on the tradition. If you want to know what jazz sounds like this very minute, you won't go wrong by giving a listen to Bill Charlap, Larry Goldings, Fred Hersch, Maria Schneider or Luciana Souza, to name five younger musicians whose work I especially admire.

Still, every list must stop somewhere, for which reason I also had to omit a flock of older players who figure prominently in my personal pantheon of excellence, starting with Bob Brookmeyer, Clifford Brown, Benny Carter, Charlie Christian, Erroll Garner, Jim Hall, Ahmad Jamal, Wes Montgomery, Red Norvo, Django Reinhardt, Shorty Rogers, Artie Shaw, Kenny Wheeler . . . but I'd probably better quit while I'm behind. So stop reading and start listening, and if you like what you hear, tear out these pages and send them to anyone who doesn't know what to give you for Christmas.

A musician who nudged jazz in more directions than any other player: Miles Davis, whose "The Complete Birth of the Cool" makes our Top 30.The man who put hard bop -- blunt, aggressive, soulful -- on the map: "Horace Silver and the Jazz Messengers" captures a pivotal jazz development.From Duke Ellington, left, to Louis Armstrong, these greats offer a listener-friendly introduction to jazz.