Policing has been a major issue. Though Ms. Walton has distanced herself from wanting to reduce police funding, Mr. Brown attacked her on the issue in a television ad.

In Boston, the runoff puts two city council members, Michelle Wu and Annissa Essaibi George, against each other, with Ms. Wu running as the progressive. Ms. Wu, who is backed by Senator Elizabeth Warren of Massachusetts, finished in first place in the primary.

In New York City, Eric Adams, the borough president of Brooklyn and a Democrat, is expected to win the mayor’s race and has already fashioned himself as a national figure. “I am the face of the new Democratic Party,” Mr. Adams declared after his June primary win.

In Miami, Mayor Francis Suarez, a rare big-city Republican mayor, is heavily favored to win re-election and is lined up to become the president of the U.S. Conference of Mayors, giving him a national platform.

And in Atlanta, a crowded field of 14 candidates, including the City Council president, Felicia Moore, is expected to lead to a runoff as former Mayor Kasim Reed attempts to make a comeback.

The future of policing is front and center

One recurring theme in municipal races is policing, as communities grapple with the “defund the police” slogan that swept the country following the police killing of George Floyd last year. The debate is raging inside the Democratic Party over how much to overhaul law enforcement — and over how to talk about such an overhaul.

Perhaps nowhere is the issue more central than in Minneapolis, the city where Mr. Floyd was killed, sparking civil unrest across the country. Voters there will decide on a measure to replace the troubled Minneapolis Police Department with a new Department of Public Safety..

Mayor Jacob Frey, who is up for re-election, has opposed that measure and pushed for a more incremental approach. His challengers, among them Sheila Nezhad, want a more aggressive approach.

Policing is a key issue not only in the Buffalo mayor’s race, but also in mayoral contests in Seattle, Atlanta and in Cleveland, where an amendment that would overhaul how the city’s police department operates is on the ballot as well.

The mayor’s race in Cleveland puts Justin Bibb, a 34-year-old political newcomer, against Kevin Kelley, the City Council president. Mr. Bibb supports the police amendment and Mr. Kelley opposes it.

House races and Pennsylvania’s court battle

There are two special elections for House races in Ohio, with Shontel Brown, a Democratic Cuyahoga County Council member, expected to win a heavily Democratic seat in Cleveland. Mike Carey, a longtime Republican coal lobbyist, is favored in a district that sprawls across a dozen counties.

Mr. Carey faces Allison Russo, a Democrat endorsed by Mr. Biden. Mr. Carey’s margin in a seat that Mr. Trump carried by more than 14 points last year will be another valuable indicator of the political environment.

In Florida, a primary is being held for the seat of Representative Alcee Hastings, who died earlier this year. The winner will be favored in a January special election.

The only statewide races happening in Pennsylvania on Tuesday are for the courts. The most closely watched contest is for the State Supreme Court, which features two appeals court judges, the Republican Kevin Brobson and the Democrat Maria McLaughlin. Democrats currently hold a 5-2 majority on the court and the seat being vacated was held by a Republican, so the result will not swing control.

But millions of dollars in advertising are pouring into the state, a sign not just of the increasing politicization of judicial contests, but also of the state’s role as a top presidential battleground.