What to Watch in Primaries in Maine, South Carolina, Nevada and North Dakota

The presidential primary season is over, but down-ballot primaries are still happening across the country. Candidates will be on the ballot on Tuesday in five states: South Carolina, Nevada, Maine, North Dakota and Ohio, where a special House race will take place.

Representative Bill Johnson retired from Ohio’s Sixth Congressional District in January, and State Senator Michael Rulli, the Republican nominee for the seat, will face off against Michael L. Kripchak, the Democratic nominee. Former President Donald J. Trump won that district in 2020 by a margin of 29 percentage points.

Here are the primaries to keep an eye on:

Nancy Mace, a Trump vice-presidential contender, will be on the ballot in South Carolina.

Representative Nancy Mace, Republican of South Carolina, who is a contender to be Mr. Trump’s running mate, faces a primary opponent backed by former Speaker Kevin McCarthy, who opposes her because she voted to oust him from the speakership last year. The primary, in the state’s First Congressional District, comes just days after the House Ethics Committee started looking into Ms. Mace’s use of a congressional reimbursement program.

Catherine Templeton, Ms. Mace’s opponent, worked for Nikki Haley when she was governor of South Carolina, and Ms. Templeton ran for that office in 2018. Ms. Mace was once a moderate ally of Ms. Haley’s, but she shifted to the MAGA wing of the Republican Party when her district was redrawn in 2022, making it more conservative.

The Third and Fourth Congressional Districts will also host competitive Republican primaries. Representative Jeff Duncan retired from his seat in the deeply conservative Third District, and, in the Fourth District, Representative William R. Timmons IV, who fended off Republican challengers in the 2022 primary, will be challenged by State Representative Adam Morgan, the leader of what is their chamber’s version of the hard-right Freedom Caucus.

The leading Republican Senate candidate in Nevada has a last-minute endorsement from Trump.

Nevada Republicans will choose their nominee for a U.S. Senate seat from a crowded field.

The leading candidate, Sam Brown, a former Army captain, received a last-minute boost in the form of an endorsement from Mr. Trump, who weighed in on the race on Sunday night, though after early voting had ended.

Still, the endorsement has solidified Mr. Brown’s standing as the heavy favorite to face Senator Jacky Rosen, the Democratic incumbent, in November. Mr. Brown also has the endorsement of the state’s Republican governor, Joe Lombardo, and he has raised the most money of any candidate in the race.

Mr. Brown will face Jeff Gunter, a wealthy dermatologist who was Mr. Trump’s ambassador to Iceland, as well as Jim Marchant, who has supported Mr. Trump’s lie that the 2020 election was stolen from him, and a number of other minor candidates.

Two North Dakota Republicans are vying to replace Gov. Doug Burgum.

Gov. Doug Burgum, Republican of North Dakota, who self-funded a failed campaign for president and who is now a leading contender to be Mr. Trump’s running mate, is not seeking a third term as governor. Two Republicans will face off on Tuesday in a bid to succeed him.

One of them, Representative Kelly Armstrong, has filled North Dakota’s lone House seat since 2019 and helped lead the House investigation of President Biden’s son Hunter. He will face Lt. Gov. Tammy Miller, who was appointed to her position by Mr. Burgum last year and who has been endorsed by the governor.

The winner will face State Senator Merrill Piepkorn, a Democrat, in November. In 2020, Mr. Burgum defeated his Democratic opponent by 42 points.

Maine Republicans will choose a challenger for Representative Jared Golden.

Representative Jared Golden, Democrat of Maine, has faced tough re-election battles in the state’s rural Second Congressional District, and Republicans will choose from two candidates who are vying to give him one more in this year’s general election.

State Representative Austin Theriault, a former NASCAR driver, will go up against State Representative Michael Soboleski. Mr. Theriault has been endorsed by Mr. Trump, who urged his supporters on the eve of the election to back him.