Update 2:25 p.m. EST Nov. 14: Bevin announced Thursday that he wouldn't contest the results of the election as recanvass totals appeared to affirm the results of last week's election, according to the Courier Journal.
He wished Beshear luck during a news conference Thursday, according to Kentucky Public Radio.
Beshear, who is the state attorney general and the son of former Kentucky Gov. Steve Beshear, will be sworn in next month, Politico reported.
Update 3:45 p.m. EST Nov. 6: Bevin formally requested a recanvass of the vote Wednesday afternoon, according to Kentucky's Secretary of State Alison L. Grimes.
A recanvass is not a recount, but rather a check of the vote count to ensure the results were added up correctly. Bevin could also seek a recount in court.
Grimes said the recanvass was scheduled to take place Nov. 14 at 9 a.m.
Original report: "Tonight, voters in Kentucky sent a message loud and clear for everyone to hear," Beshear, currently Kentucky's attorney general, said in a victory speech late Tuesday. "It's a message that says our elections don't have to be about right versus left; they are still about right versus wrong."
Beshear, whose father, Steve Beshear, served as governor from 2007 to 2015, added that he hoped Bevin "will honor the election that was held tonight," the AP reported.
>> Watch Beshear's speech here
But Bevin was not ready to give in.
"Would it be a Bevin race if it wasn't a squeaker? I mean, come on," he said. "I mean, really and truly, this is a close, close race. We are not conceding this race by any stretch."
He added that "we truly don't know" who the next governor will be.
"Whoever it is will be the one determined by the process being followed, by the law being followed, by the process being truly sound," Bevin said, claiming that there may have been unspecified "irregularities."
>> See Bevin's speech here
Democrats hailed the results as a win against President Donald Trump, who supported Bevin and appeared with him at a rally in Kentucky on Monday.
Meanwhile, Brad Parscale, Trump's campaign manager, released a statement saying the president "just about dragged Gov. Matt Bevin across the finish line."
Trump echoed the sentiment in a series of tweets early Wednesday and pointed out that Republicans won five of six statewide races.
Read more here or here.
– The Associated Press contributed to this report.