Stevenson represented his country in the 1972 Olympic Games of Munich. He won the Gold medal there, and then in the 1976 games, held in Montreal, Stevenson repeated the feat. By then, he had become a national hero in Cuba, and he was a household name in his island. This was the point where he was the closest to sign a professional contract, American fight promoters offering him the amount of five million dollars to challenge world Heavyweight champion Muhammad Ali in his first professional bout, which would have made him the second boxer to go straight from the Olympics into a professional debut with the world's Heavyweight crown on the line, after Pete Rademacher. But he refused, asking What's five million dollars worth, when I have the love of five million Cubans?. Stevenson went to the 1980 Olympics in Moscow and became the second boxer ever, after Papp, to win three Olympic boxing gold medals. He and Papp remain the only two boxers to do that feat.
Stevenson might have won a fourth gold medal at the 1984 Los Angeles Olympic Games, but Russia had boycotted those games in answer to the United States boycott of the 1980 Moscow competition, and Cuba stood by it's allied country, so he wasn't given a chance to earn gold for the fourth time. He retired from boxing shortly after.
Stevenson was named coach of Cuba's amateur boxing program, and Cuban President Fidel Castro presented him with a mansion in an exclusive residential sector, a practice that Castro is known for with Cuban athletes that have done good for their country while not defecting. In 1999, he ran into trouble at Miami International Airport when, before boarding a United Airlines chartered jet that would take the Cuban national boxing team back home, he allegedly headbutted a 41 year old United ticket counter employee, causing him to break his teeth. He was arrested, but soon after, he was let go and returned home.
When Stevenson refused to turn professional and fight Ali, there were some other good professional Heavyweights around, like Ken Norton, Larry Holmes, George Foreman, Joe Frazier, Chuck Wepner and Ron Stander. Most of those, except for Wepner and Stander, were world Heavyweight champions at one point of their careers, leaving the question of whether Stevenson would have been able to succeed as a professional or not as one of boxing's biggest unanswered questions of all times.