Mr Trebek announced he had been diagnosed with stage-four pancreatic cancer in March 2019.
The Jeopardy! Twitter account said on Sunday he had "passed away peacefully at home" surrounded by family and friends.
Mr Trebek had hosted Jeopardy! since 1984, and had received numerous awards and honours for his work.
Producer of Jeopardy!, Sony Pictures, led tributes to the "legend", writing in a statement: "For 37 amazing years, Alex Trebek was that comforting voice, that moment of escape and entertainment at the end of a long, hard day for millions of people around the world."
Known for his sharp wit and charisma, the Canadian-American presenter became the face of Jeopardy! during his three decades on the show, turning it into a ratings smash hit.
He fronted more than 8,200 episodes of the popular quiz show, making him among the most well-known people on television in the US and Canada.
In 2014 he set a Guinness World Record for "most game show episodes hosted by the same presenter".