No one thought it would be easy for Illinois voters to pick a new governor after 14 years with retiring Gov. James R. Thompson (R) -- and no one was right. The battle between Attorney General Neil Hartigan (D) and Secretary of State Jim Edgar (R) is very close. Hartigan is suffering at least minor defections in his Chicago base, where the independent Harold Washington Party is urging black voters to blank him. But Edgar has to worry about low turnout and possible slippage in suburban areas, where Hartigan's anti-tax message has hurt the Republican because of his support for continuation of a temporary surtax. Late private polls indicate if anyone has a tiny edge, it is Hartigan.

By contrast, Sen. Paul Simon (D) is comfortably out in front of his opponent, Rep. Lynn Martin (R), whose efforts to criticize Simon's liberalism and his aid to contributors were hampered by voters's perception that she was too caustic in her attacks. Martin's traditionally Republican seat in Rockford is seeing an unusually close race, with state Rep. John W. Hallock Jr. (R) at best a slight favorite over former county prosecutor John W. Cox Jr. (D). Veteran Rep. Frank Annunzio (D), whose close S&L ties made him a GOP target, is holding on in his working-class Chicago district against state Sen. Walter W. Dudycz (R), and freshman Rep. George E. Sangmeister (D) is favored over Homewood Mayor Manny Hoffman (R) in a district that belonged to the Republicans most of the 1980s. INDIANA

Sen. Daniel R. Coats (R), appointed from his House seat to succeed Vice President Quayle, took a lot of ridicule from his challenger, state Rep. Baron Hill (D), for dropping 13 million pieces of franked mail on his new constituents in one year. But the name recognition he gained has paid off and Coats is cruising against his ill-financed opponent.

The former Coats-Quayle House seat in Fort Wayne seems likely to stay with Rep. Jill Long (D), who stunned Republicans with her 1989 special-election victory and has fashioned a voting record that protects her against television preacher Rick Hawks' (R) efforts to depict her as a flaming liberal.

But anti-incumbent sentiment has surfaced strongly in two other districts. Five-term Rep. John Hiler (R) has never had an easy race in South Bend and this year former congressional aide Timothy J. Roemer (D) may strike pay dirt with his financial conflict-of-interest charges. But the GOP is upbeat about chances of beating Rep. Jim Jontz (D), who has won twice by dint of sheer hard work in strongly Republican territory. He has a tough opponent in entrepreneur John A. Johnson (R), who has plenty of money to trumpet his anti-Washington message. IOWA

Two-term Gov. Terry E. Branstad (R), who has bent some of his early conservatism to the mood of his state, apparently has put away the challenge from state House Speaker Don Avenson (D), who targeted Branstad's antiabortion views. Today's Des Moines Register poll shows Branstad leading 57 to 33 percent, but many insiders doubt the margin will be that wide.

The same poll puts Sen. Tom Harkin (D) ahead of Rep. Thomas J. Tauke (R) by 48 to 39 percent. The two men quarreled over everything from pay raises to vacation homes. Harkin is the abortion-rights supporter in this race, but economic issues probably have as much or more to do with his being in a position to break the reelection curse that has tripped other Iowa Democratic senators. An upset is unlikely but not impossible.

Tauke's Senate candidacy may cost the GOP the House seat he has held for 12 years. Lawyer-farmer Eric Tabor (D), who ran twice against Tauke, is a slight favorite over Delaware County attorney Jim Nussle (R), running for the first time in the 12-county district. KANSAS

Gov. Mike Hayden (R), 46, was so unpopular because of tax hikes that he barely survived the GOP primary and seemed certain to fail in his bid for a second term. But state Treasurer Joan Finney, 65, surprise winner of the Democratic nomination, has proved to be a disastrous candidate. Not only has her strict antiabortion stance moved one-third of the Democratic vote to Hayden, but her gaffes have pushed anti-Hayden voters and newspapers into giving him grudging support. He figures to slip by in this hold-your-nose contest.

Sen. Nancy Landon Kassebaum (R) has a walkover for her third term against political unknown Dick Williams (D). Banker Dick Nichols (R) should have no trouble against rancher George Wingert (D) to succeed retiring Rep. Bob Whittaker (R), but redistricting may make him a one-termer. MICHIGAN

Knowing it would be no cinch to win a third term in a slumping auto economy, Gov. James J. Blanchard (D) went on the attack early against his challenger, state Senate Majority Leader John Engler (R). The tactic apparently has worked. While Engler has closed the margin to a handful of points in private polls, his negatives -- and his antiabortion stance -- are causing enough doubts, especially among suburban Republican women, to keep him the underdog.

Sen. Carl M. Levin (D) used a different but even more effective strategy against his challenger, Rep. Bill Schuette (R). He built a huge campaign chest against Schuette's family fortune, then rebutted every criticism Schuette made of his liberal record. The likely result: a handsome win.

Schuette has handed on his House seat to a former aide, state Rep. David Camp (R), whose opponent is political novice Joan Dennison (D). A woman will be joining the Michigan delegation, however, when Barbara-Rose Collins (D), a city council ally of Detroit Mayor Coleman Young, takes over from retiring Rep. George W. Crockett Jr. (D). Carl Edwards is the token Republican opponent. MINNESOTA

In this normally staid Scandinavian state, politics has taken a walk on the wild side. The governor's race was thrown into turmoil in mid-October when three women alleged that, as teenagers in 1981, they had gone skinny-dipping with Independent-Republican nominee Jon Grunseth. A couple of turns in the plot later, he dropped out of the race -- just nine days before the election. He's been replaced on the ballot by the man he defeated in the primary, state Auditor Arne Carlson, a moderate Republican who supports abortion rights.

Ten-year incumbent Gov. Rudy Perpich (DFL), a foe of abortion, may find Carlson, who has won two statewide campaigns, a more formidable foe than Grunseth. Polls published today by the St. Paul Pioneer Press and the Minneapolis Star Tribune showed the two running dead even, with a huge 22 percent of the electorate undecided.

Two-term GOP Sen. Rudy Boschwitz's big fear is that Republican voters turned off by the antics in the governor's race will stay home on Tuesday and deprive him of an expected win over Democratic-Farmer-Labor Senate nominee Paul Wellstone, a college professor and community organizer who has tried to turn Boschwitz's 7 to 1 funding advantage against him. Late polls show Boschwitz in the lead.

The most imperiled Republican in the state is Rep. Arlan Stangeland, who is in hot water with voters for making hundreds of telephone calls at taxpayer expense to a female lobbyist. He is being challenged by former state senator Collin C. Peterson, who has picked up many political scars in four previous tries for the seat. MISSOURI

If there is an anti-incumbent trend, Missouri voters can be more contrary than a mule in resisting it. All nine of their representatives have return tickets to the Capitol that they are almost certain to be able to use. NEBRASKA

Like Republican Gov. Mike Hayden in neighboring Kansas, Gov. Kay A. Orr (R) found herself the target of a tax revolt and, like him, she may have found political salvation in her opponent, wealthy businessman Ben Nelson (D), who had to endure a lengthy primary recount to gain the nomination. Her ads have raised enough doubts about Nelson's financial ties to special interests that she has erased his early lead and may squeak through to a second term.

Sen. J. James Exon (D), with a conservative voting record and the clout of his Armed Services Committee seniority, is far out front of his third-term challenger, former representative Hal Daub (R). Daub was hurt by the backlash to former Texas senator John G. Tower's score-settling remark, while in Omaha, that Exon was notorious for his drinking habits.

Democrats have hopes that Sandra K. Scofield (D), a teacher and farmer, can win the huge western district seat being vacated by Rep. Virginia Smith (R). Her opponent, Bill Barrett (R), is the speaker of the unicameral legislature and bears some scars from the tax fight. The district is heavily Republican, but is accustomed to having a woman in Congress and Exon's coattails may help Scofield. NORTH DAKOTA

The Lawrence Welk homestead controversy -- a $500,000 appropriation to preserve the bandleader's birthplace in Strasburg, N.D. -- has turned into a campaign ad for Edward T. Schafer (R), the detergent heir challenging Rep. Byron L. Dorgan (D). In a dull campaign year, that's about all people talk about. But don't look for Dorgan on the congressional casualty list. OHIO

The Buckeye State was picked by Republicans as their main takeover target in this year's gubernatorial campaigns -- and it looks as if they'll succeed. Former Cleveland mayor George Voinovich (R) has the edge on state Attorney General Anthony Celebrezze (D). The Democrat made some headway by pushing his new-found abortion-rights stance, but Voinovich's strength in the Democratic base in Cleveland and a time-for-a-change feeling after eight years of retiring Democratic Gov. Richard F. Celeste's administration give the GOP the edge.

That advantage may not reach to the congressional battle in Cincinnati's 1st District, where Mayor Charles Luken (D), son of retiring Rep. Thomas A. Luken (D), faces former mayor J. Kenneth Blackwell (R). Blackwell, probably the strongest of three GOP black congressional candidates around the country, has been drawing only half the black vote in polls, but may have enough of a grass-roots organization to offset Luken's TV edge. The Democrat is a slight favorite.

The GOP should have little difficulty electing new members in two open seats it now holds. State Sen. David L. Hobson (R) is slated to succeed Rep. Michael DeWine (R), Voinovich's running mate, and state Rep. John A. Boehner (R) is moving in to replace Rep. Donald E. "Buz" Lukens (R), who resigned in the face of sex scandals. SOUTH DAKOTA

Sen. Larry Pressler (R) is a classic example of a politician whose standing with his constituents is much higher than with his colleagues. A maverick with prickly relations even within his own party, Pressler drew a serious challenge from Ted Muenster (D), a businessman and college administrator who has strong support from other Democrats. But despite negative publicity in the state's largest paper for examples of lavish living in Washington, Pressler retains the goodwill of his largely rural constituency. South Dakota is a state that likes to reelect incumbents and Muenster has lacked the money to force voters to confront that habit. WISCONSIN

Gov. Tommy G. Thompson (R) could give other Republicans political lessons, so skillfully has he entrenched himself as a conservative in this liberal-leaning state. The breadth of his appeal -- from Republican dairy farms to Milwaukee streets -- has left Assembly Speaker Tom Loftus (D), an articulate liberal, far in arrears.