NEW YORK — A musical about chance meetings in the Israeli desert and a set of plays about magical happenings in a wizarding universe claimed the top honors Sunday night at the 72nd annual Tony Awards.

“The Band’s Visit,” a seductively atmospheric 90-minute work with an alluring score by David Yazbek, and “Harry Potter and the Cursed Child,” a 5½ -hour stage addition to J.K. Rowling’s mega-successful fantasy franchise, collected the statuettes for best musical and best play, respectively, in a three-hour ceremony from New York’s Radio City Music Hall, most of whose results were never much in doubt.

Winning in a category that included three musicals derived from better-known pop-cultural sources — “Frozen,” “Mean Girls” and “SpongeBob SquarePants” — “The Band’s Visit” scooped up a total of 10 awards, including those for best book (Itamar Moses), score, sound, orchestrations, lighting, direction (David Cromer), featured actor (Ari’el Stachel) and actor and actress in a musical (Tony Shalhoub and Katrina Lenk); “The Producers’s” record 12 wins in 2001 still stands. “Harry Potter,” meantime, swept the design categories for plays but got no acting recognition, earning six trophies as it won for direction, set, sound, costumes and lighting.

The British actress Glenda Jackson, returning to Broadway for the first time in 30 years, was the winner for best actress in a play for infusing a dying dowager with the energy of roaring rapids in Edward Albee’s “Three Tall Women.”

“I think Joe Mantello found me a worthy opponent,” Jackson joked about her director. Of the reception she has gotten in New York, she added: “You, as always, are welcoming and kind and generous. And America has never needed that more. And America is always great.”

Tina Fey, another celebrated fixture of the 2017-2018 Broadway season for her book for “Mean Girls,” based on her 2004 movie, lost in a mild upset to Moses, and her show went away completely empty-handed. So did “Frozen,” another show popular with younger theatergoers.

Best revival of a play went to an epic, London-born restaging of Tony Kushner’s two-part “Angels in America,” which also earned awards for lead actor Andrew Garfield, as AIDS-stricken Prior Walter, and for supporting actor Nathan Lane, playing (against type) as thuggish legal fixer Roy Cohn. And in a win regarded as come-from-behind, “Once on This Island” bested Lincoln Center Theater’s “My Fair Lady” and “Carousel” for best musical revival.

Garfield dedicated his award to gay, lesbian and transgender people, and alluded to the recent U.S. Supreme Court decision upholding the right of a Colorado baker to deny his services to a gay couple. “Let’s just bake a cake for everyone who wants a cake to be baked,” he said. And Lane choked up as he spoke of winning his third Tony for a dramatic, rather than a comic performance. “This award is a lovely vote of confidence that I’ve been on the right path,” he said.

Hosting duties were shared by a pair of artists representing an attempt to broaden Broadway’s fan base: singer Josh Groban, a big audience draw during the previous season when he appeared in the musical “Natasha, Pierre and the Great Comet of 1812,” and singer-songwriter Sara Bareilles, whose musical “Waitress” is a solid Broadway crowd-pleaser. (Last year, you might recall, the evening’s host was Kevin Spacey, later accused of sexual misconduct.)

But the event’s most coveted newcomer to Broadway — from the standpoint, certainly, of a potential TV ratings spike — was Bruce Springsteen, whose “Springsteen on Broadway” has been a box-office bonanza all season long at the Walter Kerr Theatre.

While Springsteen declined to participate in any competitive Tony categories — which would have required his show to allot pairs of tickets to the more than 800 Tony voters — he was still awarded a special Tony on Sunday night, and performed live in Radio City.

Introduced by Robert DeNiro, who whipped up the audience with an expletive about the president that had to be bleeped, Springsteen offered a monologue from his show and sang “My Hometown.”

Groban and Bareilles were the kind of affable emcees who remind you that making witty repartee and introducing other stars is such a workmanlike assignment that the best you hope for is the proceedings don’t feel as if they run longer than Wagner’s Ring Cycle. This 72nd incarnation included among its presenters “Hamilton” stars Leslie Odom Jr. and Christopher Jackson; CBS stars Jim Parsons and Christine Baranski; venerable stars Mikhail Baryshnikov and DeNiro; Broadway stars Bernadette Peters and Patti LuPone; and TV ­streaming-service stars Uzo Aduba and Rachel Brosnahan.

The co-hosts sat at facing grand pianos for the opening number, a low-key salute to all the nominees who lose, a well-sung (naturally) ditty that was a counterpoint to the usual boffo production number with which the Tonys show likes to start. What ensued was a pleasant if unexciting program during which numbers from the nominated musicals were trotted out in dutiful fashion. And then about 75 minutes in, actor Matthew Morrison introduced a song that brought the audience tearfully to its feet: Students from the drama department of Marjory Stoneman Douglas High School in Parkland, Fla., where 17 students and school employees were murdered earlier this year in a shooting spree, performed quite a fine rendition of “Seasons of Love” from “Rent.”

An awards show doing not only a smart thing, but also the smartest thing: That’s an achievement.

The wins by “The Band’s Visit” and “Harry Potter” — during a Broadway season generally thought to be of middling quality — spoke to the tension in the dueling priorities of Broadway these days, between riskier artistic endeavors and brand-name theatrical products with presold audience bases. “The Band’s Visit” was, like the other musicals in its category, adapted from other material. But that source was a little-known Israeli indie movie about a visiting Egyptian military band ending up in the wrong Israeli town, rather than a kid-oriented movie or TV series. The Tony voters rewarded it for its daring.

The British import “Harry Potter,” on the other hand, regardless of how beautifully crafted by director John Tiffany and a passel of creative collaborators, performs at the box office more like a hit musical than a play and, as such, in no way advances the cause of plays and playwriting on Broadway. That its competition in the category was so lackluster was one of the season’s biggest letdowns, especially after the much finer crop of plays that contended in the 2016-2017 Tonys, in which J.T. Rogers’s “Oslo” came out on top.

In some other notable results, Melody Herzfeld, drama teacher at Marjory Stoneman Douglas, received the Tonys’ excellence in theater education award. And Laurie Metcalf, who recently lost the Oscar for “Lady Bird,” won her second Tony in as many years, this time for playing a younger version of Glenda Jackson’s character in “Three Tall Women.”

Highlights from the 2018 Tony Awards

Gavin Lee from “SpongeBob SquarePants: The Musical” performs. (Theo Wargo/(credit Too Long, See Caption))