What to watch in Tuesday’s primaries.

Voters in five states head to the primaries on Tuesday to decide races that will shape the Republican Party and perhaps America’s democratic future this November and beyond, with former President Donald J. Trump playing key roles in marquee races in Arizona, Michigan and Washington.

Few states have been more rattled by Mr. Trump’s baseless claims of election rigging than Arizona and Michigan. On Tuesday, Republican voters in those states will choose standard-bearers for governors’ races in November, and in Arizona will also nominate a candidate for secretary of state, the post that oversees elections.

Also on the ballot will be the Republican nominations for Senate races in Arizona, Missouri and Washington. Republican voters will also decide the fate of three of the 10 House Republicans who voted to impeach Mr. Trump for inciting the Jan. 6, 2021, attack on the Capitol.

Here are the key races to watch:

In Arizona, Trump is front and center.

The former president turned against Arizona’s governor, Doug Ducey, after Mr. Ducey certified Joseph R. Biden Jr.’s narrow victory in the state and refused to echo Mr. Trump’s lies about a stolen election. The race to succeed Mr. Ducey has been dominated by that issue.

Mr. Trump’s preferred candidate, the former news anchor Kari Lake, has repeated outlandish falsehoods about the 2020 election and embraced provocations like vowing to bomb smuggling tunnels on the southern border. Her main opponent, Karrin Taylor Robson, a real estate developer endorsed by Mr. Trump’s vice president, Mike Pence, is running on conservative themes but not on election denial.

On the Democratic side, Katie Hobbs, Arizona’s secretary of state, is favored to win the nomination, setting up what is expected to be a tight, high-stakes contest this fall.

Mr. Trump again figures in the Republican primary to take on Senator Mark Kelly, a Democrat, this November, a key front in the battle for control of a Senate now divided evenly between the parties. The former president’s endorsement of the political newcomer Blake Masters helped vault the quirky technology executive into the lead, but the state’s attorney general, Mark Brnovich, could benefit from the barrage of attack ads aimed at Mr. Masters from another Senate candidate, the solar power executive Jim Lamon.

The race for the Republican nomination for secretary of state features Mark Finchem, a state representative and expansive conspiracy theorist who marched on the Capitol on Jan. 6, 2021.

Show me the fate of Eric Greitens.

The race to succeed Senator Roy Blunt, the Missouri Republican who is retiring, should have been a gimme for the Show Me State’s Republicans, who now dominate statewide office. But the attempted political comeback of Eric Greitens has complicated matters. In 2018, Mr. Greitens resigned as governor in disgrace amid an investigation into fund-raising improprieties and an allegation by his former hairdresser that he had lured her to his home, stripped off her clothes, taped her to exercise equipment, photographed her, threatened to make the photos public if she talked and then coerced her into oral sex.

Taking a page from Mr. Trump, Mr. Greitens dismissed the allegations as cooked up by his political enemies — Democrats and “Republicans in name only” — as he plotted a comeback by running for Senate. Prominent Republicans in Missouri and Washington, D.C., split their endorsements between the state’s attorney general, Eric Schmitt, and a conservative House member, Vicky Hartzler, giving Mr. Greitens a path to the nomination — and Democrats a plausible shot at the seat.

In the closing weeks, affluent donors dumped money into an anti-Greitens super PAC, Show Me Values, which blistered Mr. Greitens with his former wife’s accusations of domestic violence against her and one of their young sons. The group’s backers were confident another candidate would prevail.

Despite Donald Trump Jr.’s backing of Mr. Greitens, his father, the former president, never came through with an endorsement.

Missouri Democrats will have a difficult time grabbing the seat even if Mr. Greitens prevails. And a new complication has threatened Democratic unity: The party had largely gotten behind Lucas Kunce, a telegenic former Marine, but his coronation was interrupted by the late rise of Trudy Busch Valentine, the free-spending heiress to the Anheuser-Busch fortune.

In Michigan, democracy (and Israel) on the ballot.

Up and down the state’s primary tickets, Michiganders who deny President Biden’s clear, 2.8-percentage-point victory in their state are vying to defeat politicians from both parties who accept the results.

Ryan Kelley, who was arrested last month by the F.B.I. for his actions at the Capitol on Jan. 6, 2021, is running to unseat Gov. Gretchen Whitmer, a Democrat, though in the most recent polling he trails the conservative media personality Tudor Dixon — whose views on the 2020 election have wavered — and the self-funding businessman Kevin Rinke.

Ms. Dixon picked up Mr. Trump’s endorsement on Friday, but it was unclear whether his supporters in the state would rally behind her after warring for months with Ms. Dixon’s chief backer, Betsy DeVos, and her relatives, the most influential Republican family in Michigan.

In the Western Michigan House seat centered in Grand Rapids, a Trump-backed election denier, John Gibbs, is trying to take out Representative Peter Meijer, a freshman Republican who not only accepts the election results but also voted to impeach Mr. Trump for inciting the attack on the Capitol.

The Democratic Congressional Campaign Committee aired an advertisement in the final days of the campaign lifting Mr. Gibbs, a potentially far weaker candidate in November than Mr. Meijer, by highlighting his conservative credentials for Republican primary voters, a move that infuriated some Democrats.

In Eastern Michigan’s Detroit suburbs, redistricting pitted two incumbent Democratic House members, Andy Levin and Haley Stevens, against one another. That race has turned into a battle royal between progressive groups backing Mr. Levin and pro-Israel groups determined to punish him for what they see as a bias toward Palestinians.

The impeachers’ penultimate stand.

Three of the 10 House Republicans who voted to impeach Mr. Trump face their day of reckoning on Tuesday. Their fate will say much about Mr. Trump’s power with primary voters. Besides Mr. Meijer, Representatives Jaime Herrera Beutler and Dan Newhouse, both of Washington, are being challenged by Republicans endorsed by Mr. Trump as part of his revenge tour.

Mr. Newhouse has largely kept his head down since voting to impeach, but he, too, has a Trump-backed challenger, Loren Culp, a retired law enforcement officer who was the Republicans’ candidate for governor of Washington in 2020.

Of the impeachment 10, so far four have retired; one, Representative Tom Rice of South Carolina, has lost his primary; and one, Representative David Valadao of California, has survived his primary. After Tuesday, just one more awaits a primary: Representative Liz Cheney of Wyoming, whose uphill fight will be decided on Aug. 16.

Abortion on the ballot.

Voters in Kansas will be the first since the Supreme Court overturned Roe v. Wade to decide for themselves whether to protect reproductive rights or turn the issue of abortion over to state legislators.

Tuesday’s ballot will include an amendment to the state constitution that would remove an existing guarantee of reproductive rights and allow the Legislature to pass laws restricting abortion.

The returns in Kansas will be closely watched, not only by abortion rights supporters and Democrats, for signs of the potency of the issue in the midterm elections, but also by Republican state lawmakers in Kansas and beyond, who felt empowered by the Supreme Court’s decision but are unsure how far they should go to bar abortion in their states.

Incumbent Democrats see danger ahead.

The power of incumbency is proved time and again, but with inflation at a 40-year-high, President Biden’s approval ratings well below 40 percent and congressional redistricting taking a toll, holding elective office is no guarantee of keeping it.

In Kansas, Laura Kelly, a Democratic governor in a deep-red state, has an approval rating of 56 percent, 23 percentage points higher than Mr. Biden’s, but her relative success may not save her tossup race against her expected Republican challenger, Attorney General Derek Schmidt.

In the Kansas City, Kan., suburbs, Representative Sharice Davids — a gay former mixed-martial arts fighter and one of the first two Native American women in the House — was hailed as a path-breaker after her 2018 victory. But redistricting redrew her seat from a slight Democratic lean to a slight Republican edge.

If Amanda Adkins, a businesswoman and former congressional aide, wins the Republican primary on Tuesday, November’s race will be a rematch of their 2020 contest, which Ms. Davids won easily. But this time, the circumstances will be more difficult for the incumbent.

If the political environment deteriorates further for Democrats, another incumbent in a Tuesday primary, Senator Patty Murray of Washington, could pop up on both parties’ radar screens.

In the nonpartisan Washington primary, Ms. Murray is expected to cruise, as will the Republican backed by the party establishment, Tiffany Smiley. A nurse and motivational speaker, Ms. Smiley will lean on a biography that includes the blinding of her husband by a suicide bomber in Iraq, a tragedy that drove her to veterans’ causes. But her main argument is that 30 years after Ms. Murray first won her Senate seat as a “mom in tennis shoes,” it’s time for “a new mom in town.”