Stevenson represented his country in the 1972 Olympic Games of Munich. He won the Gold medal, and then in the 1976 games, held in Montreal, Stevenson repeated the feat. By then, he had become a national hero in Cuba, and was a household name in Cuba. 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. At the 2000 Summer Olympics, Félix Savón, also from Cuba, became the third boxer to achieve this.
Stevenson might have won a fourth gold medal at the 1984 Los Angeles Olympic Games, but the Soviet Union boycotted the games in retaliation for the United States boycott of the 1980 Moscow competition. Cuba followed the Soviet's lead, and Stevenson was deprived of the chance to earn a fourth gold. 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 area, a practice that Castro is known for. 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 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 released and returned home.
When Stevenson refused to turn professional and fight Ali, the heavyweight scene was vibrant, with fighters of the calibre of Ken Norton, Larry Holmes, George Foreman and Joe Frazier competing. Stevenson would certainly have stirred up the professional boxing world, and fight fans continue to debate the possible outcomes had he fought in the halcyon days of heavyweight boxing.