The Canadian-born presenter passed away surrounded by friends and family at his California home, a spokesperson said.
Among those to offer their condolences was Canadian prime minister Justin Trudeau, who said "we have lost an icon".
"Almost every night for more than three decades, Alex Trebek entertained and educated millions around the world, instilling in so many of us a love for trivia," he added.
"My deepest condolences to his family, friends, and all who are mourning this tremendous loss."
A long list of TV stars, singers, broadcasters and fans also paid their respects to Trebek, who was diagnosed with cancer in March 2019.
"Part of our lives for decades, he taught us about so many things and now Alex Trebek has gone from us," said Nancy Sinatra, sister of Frank.
"Sending love to his family and friends. We will miss you, sir. Godspeed," she added.
Call Me by Your Name and The Social Network actor, Armie Hammer, said: "Greatest host of one of the greatest shows of all time who passed away sadly after a long battle with cancer."
Trebek became a recognisable face across the country after spending almost four decades hosting Jeopardy!, which dubs itself as "America's favorite quiz show".
He vowed to continue presenting the show despite becoming becoming ill and often gave fans an update on his condition.
Access unlimited streaming of movies and TV shows with Amazon Prime Video Sign up now for a 30-day free trialSign up
"RIP Alex Trebek. A true, true gentleman and courageous hero," said the Academy-Award winning actress, Viola Davis. "May flights of angels sing thee to thy rest".
"My heart is heavy hearing of Alex Trebek’s passing," said singer Charlie Puth. "The soundtrack of my life was hearing his voice every night at 7pm."
He added: "There will never be a host quite like him ever again. God bless him and his family. And may his legacy forever live on."
"Jeopardy!" debuted on NBC in 1964 with Art Fleming as emcee and was an immediate hit. It lasted until 1975, then was revived in syndication with Trebek.
Trebek was more than qualified for the job, having started his game show career on "Reach for the Top" in his home country.
The show made him famous. He won five Emmys as its host and received stars on both the Hollywood and Canadian walks of fame. In 2012, the show won a prestigious Peabody Award.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies