He is best known for reaching the men's singles final at Wimbledon in 1962, where he was defeated by fellow Australian Rod Laver.
In 1958 he won the Boys' Singles title at the Australian Championships, as well as the Boys' Doubles (w/Bob Hewitt).
He was runner-up in the Men's Doubles at the Australian Championships in 1961. In 1962 he was in the finals of the Dutch Open in Hilversum, and he won the men's singles title at the Italian Championships three times in 1963, 1965 and 1967. He won singles titles in 1966 and 1967 at the Swedish Open in Bastad and the Austrian Open at Kitzbuhel. Mulligan won the 1970 Japan Championships. He was ranked in the world's top 10 in 1962, '63, '65 and '67, reaching as high as World No. 4 in the latter year. Mulligan won 1960 U.S. Claycourts doubles (w/Hewitt) and the 1962 German Doubles (w/Hewitt). Mulligan was a member of the 1968 Italian Davis Cup team, playing a total of eleven matches. Mulligan was ranked No 1 in Italy between 1968 and 1971. He was the coach of Italian Davis Cup team for ten years and was the first non-Italian to receive the Golden Racquet Award by the Italian Tennis Federation.
In the Open-era, he won only 20 tournaments, mainly on clay. The last in 1973 in Aix-en-Provence against the Argentine Julian Ganzabal. Regarding the Grand Slam tournaments, he reached a single quarter-final at Roland Garros in 1970.
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