Gov. Andrew M. Cuomo had a decisive win against Cynthia Nixon on Thursday in the Democratic primary in New York, the only state in the nation that held separate state and federal primary elections in this tumultuous political year. And though it was status quo at the top, with the lieutenant governor Kathy Hochul also prevailing, other challengers benefited from the progressive wave sweeping national politics. Here’s a look at some critical races: Read more

Cynthia Nixon, the actress and education activist, had attracted attention for her candidacy against Gov. Andrew M. Cuomo, running an aggressive campaign that some have credited for pushing Mr. Cuomo to the left. Her challenge, which, according to public polls taken in the race, seemed to be insurmountable: The governor had the benefits of incumbency, the state Democratic Party apparatus and an overwhelming fund-raising advantage. He had spent more than $21.4 million and was pouring nearly $500,000 per day into the race as of late August.

The sudden resignation of Eric T. Schneiderman, after he was accused of violence against several romantic partners, left a vacancy that four Democrats were vying to fill. Polls suggested that three of the four candidates — Representative Sean Patrick Maloney; Letitia James, the New York City public advocate; and Zephyr Teachout, a law professor — had the best chance of representing the party in November. Ms. James would be the first black woman elected to statewide office in New York, as would Leecia Eve, a Verizon executive and a former aide to Hillary Clinton.

Lt. Gov. Kathy Hochul, Mr. Cuomo’s running mate, was fending off an energetic challenge from Jumaane D. Williams, a city councilman from Brooklyn. The race is normally an afterthought to the governor’s race, but polls showed Ms. Hochul, who won the post in 2014, leading Mr. Williams by a much smaller margin than Mr. Cuomo leads Ms. Nixon.

In the State Senate, a number of races were being closely watched, including those involving incumbents who had belonged to the so-called Independent Democratic Conference, a group of breakaway Democratic senators who had had a power-sharing agreement with Republicans. Under pressure from Mr. Cuomo, the group disbanded in April and returned to the chamber’s mainstream Democrats. Still, dozens of anti-Trump resistance groups backed a slate of candidates who were mounting energetic challenges to the renegade senators, arguing that their deal with Republicans had prevented a raft of progressive bills from becoming law.

Read more about today’s most important races.