He was the chess columnist from 1972 to 2006 for The New York Times, which ran his final column (a recounting of his 1952 victory over David Bronstein) on November 12, 2006.

They were part of a talented new generation of young American masters that also included Larry Evans, Arthur Bisguier, and George Kramer.

Robert Byrne's first Master event was Ventnor City in 1945, where he scored a respectable 4/9 to place 8th; the winner was Weaver Adams.

In the 1951 Maurice Wertheim Memorial in New York, Byrne scored 6/11, tying for 6th–7th place; this was a Grandmaster round-robin featuring six of the world's top 36 players, and was won by Samuel Reshevsky.

Byrne shared 4th–7th positions at the 1957 U.S. Open Chess Championship in Cleveland with 9/12, a point behind joint winners Bobby Fischer and Arthur Bisguier.

In 1960, Byrne increased his serious play, winning the U.S. Open Chess Championship at St. Louis, and taking a silver medal on third board at the Olympiad in Leipzig.

On that same South American trip, he dominated a small but strong event at Santa Fe with 6½/7, ahead of Miroslav Filip, Aleksandar Matanović, and Héctor Rossetto.

In 1964, Byrne's third-place finish at the Buenos Aires tournament (behind Paul Keres and World Champion Tigran Petrosian), with 11½/17, made him an International Grandmaster.

Subsequently, scoring 12½/17, Byrne achieved his career highlight of third place at the Leningrad Interzonal in 1973, which made him only the fourth American (after Reshevsky, Bobby Fischer, and Pal Benko) to qualify for the Candidates Tournament (part of the world chess championship process).

However, he lost his first-round Candidates match to former world champion Boris Spassky by 1½–4½ at San Juan, Puerto Rico, in 1974.

As a 1974 Candidate, Byrne was seeded directly to the 1976 Biel Interzonal, where he performed very strongly but missed a playoff berth by half a point, sharing 5th–6th places with 11½/19, behind Bent Larsen (winner, with 12½), Mikhail Tal, Lajos Portisch, and Petrosian (2th–4th, with 12).