Here are results from key contests in Georgia, Alabama, Arkansas and Texas.

Ever since former President Donald J. Trump lost in the state of Georgia during the 2020 presidential election, he has sought revenge against the Republican incumbents there whom he blamed for not helping him overturn the results.

On Tuesday, Mr. Trump lost in Georgia again, with his endorsed candidates losing in their Republican primaries for governor, secretary of state and attorney general.

But those weren’t the only races that voters decided on Tuesday. Here is a rundown of the winners and losers in some of the most important contests in Georgia, Alabama, Arkansas and Texas:

Georgia’s secretary of state won his primary, despite Trump’s best efforts.

Secretary of State Brad Raffensperger defeated Representative Jody Hice, the Trump-endorsed candidate who had embraced Mr. Trump’s lie about widespread voter fraud during the 2020 election.

Mr. Trump targeted Mr. Raffensperger after he resisted the former president’s call to “find” additional votes after his 2020 loss in the state. Mr. Trump’s involvement in the primary fight elevated a normally overlooked contest into one of the nation’s highest profile races so far this year.

Mr. Raffensperger, who pitched himself to voters as a champion of “election integrity,” will face the winner of the Democratic primary this fall. That primary is headed to a runoff. Bee Nguyen, a member of the Georgia General Assembly, was the top vote-getter but didn’t clear the 50 percent threshold, and will face either Dee Dawkins-Haigler, a former state representative, or Michael Owens, a cybersecurity expert and former Marine.

The Georgia governor who stood up to Trump easily defeated a Trump-backed challenger.

Gov. Brian Kemp trounced David Perdue, the former president’s handpicked candidate, to win Georgia’s Republican nomination for governor. Mr. Kemp will face Stacey Abrams, the Democratic nominee, whom he narrowly defeated four years ago.

Mr. Kemp infuriated Mr. Trump by not supporting his efforts to overturn the state’s election results in 2020. Mr. Trump then recruited a series of candidates throughout the state to challenge lawmakers like Mr. Kemp.

It is perhaps the biggest loss yet for Mr. Trump in his war against Republicans who do not support his election lie. While Mr. Perdue embraced Mr. Trump’s falsehoods, Mr. Kemp did not, and used the power of his office to focus on issues like guns and the economy to appeal to conservative voters.

Chris Carr, the state’s attorney general, also defeated his Trump-backed challenger, John Gordon, to win the Republican nomination for that office. Mr. Gordon had also embraced Mr. Trump’s election lie and made that a key part of his appeal to voters.

Herschel Walker will face Senator Raphael Warnock in November.

Herschel Walker, the former football star and a Trump-backed Senate candidate, defeated a crowded field of challengers. Mr. Walker’s celebrity, fund-raising prowess and support from the former president, helped put distance between himself and his lesser-known challengers.

It also helped him brush aside questions about false statements he made about his business background, and accusations that he abused his former partner. Mr. Walker will now face Senator Raphael Warnock, who was elected to the seat in a high-profile contest last year and has raised record amounts of money heading into the general election.

Mr. Warnock’s victory last year helped give Democrats control of that chamber, and Republicans are eager to win the seat back.

In Georgia, one House Democrat beat another House Democrat in a primary orchestrated by Republicans.

Representative Lucy McBath defeated Representative Carolyn Bourdeaux and won the Democratic nomination for Georgia’s Seventh Congressional District.

It was a primary battle put in motion by a Republican-controlled redistricting process. After Republican lawmakers removed a chunk of Democratic voters from Ms. McBath’s district, she decided to run in the neighboring district, where Representative Carolyn Bourdeaux was also seeking re-election.

Ms. McBath is a Black woman from the suburbs of Atlanta and has been embraced by several liberal organizations and some progressives like Senator Elizabeth Warren of Massachusetts. Ms. Bourdeaux, a white moderate, has been skilled at appealing to those in historically center-right territory.

Marjorie Taylor Greene won her Georgia House primary.

Representative Marjorie Taylor Greene, a first-term Republican known for her far right beliefs, defeated a crowded field of challengers to win the Republican nomination in her district. Ms. Greene was stripped of two committee assignments last year for endorsing violence against Democrats, and spreading dangerous and bigoted misinformation.

Some Republicans in her district questioned whether voters would rebel, as they did last week in pushing out Madison Cawthorn, another first-term, far right House member who had been engulfed in scandals of a different sort. They did not.

In Texas, a scandal-scarred attorney general defeats a challenger named Bush.

Attorney General Ken Paxton of Texas defeated George P. Bush to win the Republican nomination for that office.

Mr. Paxton has embraced Mr. Trump’s lies about the 2020 election and was relying on Mr. Trump’s endorsement to help shield him from a host of scandals. He has been indicted on criminal securities-fraud charges and several of his top aides claimed he abused his office by helping a wealthy donor. Mr. Paxton has also faced abuse-of-power and bribery accusations.

Mr. Bush tried to align himself with Mr. Trump, distancing himself from the Bush political dynasty, which had largely united in opposition to Mr. Trump. His defeat may signal that the formerly powerful Bush family name has become a political liability in the state.

Sarah Huckabee Sanders won the Republican nomination for governor of Arkansas.

Sarah Huckabee Sanders, a former White House press secretary under Mr. Trump and the daughter of former Gov. Mike Huckabee of Arkansas, easily won the Republican nomination for governor in the state. Ms. Sanders defeated Doc Washburn, a local conservative talk radio host who was fired after he chose not to comply with his station’s Covid-19 vaccine requirement.

Mo Brooks made it into an Alabama Senate runoff after Trump pulled back endorsement.

Mr. Trump turned on Representative Mo Brooks as he slid in the polls earlier this year, but Mr. Brooks regained enough traction with voters to qualify for the run-off with the top vote-getter in the race,

Katie Britt, a former top aide to retiring Senator Richard Shelby, whom they hope to replace.

Ms. Britt, Mr. Brooks and a third leading candidate, Mike Durant, a military contractor, did not agree to debate, so voter impressions were shaped to an unusual degree by ads, many of which were funded by super-PACs and outside groups

In Texas, a Democratic House run-off was too close to call.

The race between Representative Henry Cuellar, a Democrat who opposes abortion rights, and his progressive challenger, Jessica Cisneros, an immigration attorney, appeared too close to call as of 1 a.m. Eastern time on Wednesday (results are being updated in real-time here).

The winner will run in November against the Republican nominee, Cassy Garcia, who worked as a top aide to Senator Ted Cruz.

In the Democratic run-off, immigration and abortion rights were central issues. Ms. Cisneros positioned herself as a supporter of abortion rights and criticized the incumbent as the last anti-abortion Democrat in congress. Mr. Cuellar ran ads calling Ms. Cisneros’s position on border security extreme and potentially harmful to residents in this border district.