It is thought that the young Szabó studied under Géza Maróczy, then a patriarchal figure in Hungarian chess who had previously trained future world champions, Max Euwe and Vera Menchik.

Prior to World War II, there were other successes, including a gold medal at the 3rd unofficial Chess Olympiad in 1936 as part of the Hungarian team and outright victory at Hastings 1938/39 (a tournament he was to hold a long association with).

He finished fifth at Groningen 1946, a tournament which included Mikhail Botvinnik, Max Euwe, Vasily Smyslov, Miguel Najdorf, Isaac Boleslavsky and Alexander Kotov.

It was at his third and final Candidates, held in Amsterdam in 1956, that Szabó made his most promising bid for a World Championship title challenge.

Into the 1960s and 1970s, he continued to excel in international competition; first at Zagreb 1964, first at Budapest 1965 (with Lev Polugaevsky and Mark Taimanov), first at Sarajevo 1972, first at Hilversum 1973 (with Geller) and tied for first at Hastings 1973/74 (with Gennady Kuzmin, Jan Timman and Mikhail Tal).