Grau played in many Argentine championships winning in 1926, 1927, 1928, 1934, 1935 and 1938 and he played for Argentina in five Olympiads; 1924, 1927, 1928, 1935, and 1939. Over the course of his career he defeated Euwe, Reti, Colle, Fine and secured draws against Alekhine, Keres and Capablanca.
Grau was captain of several Argentine teams. Also he was founder of the Buenos Aires Chess Circle and Argentina Chess Federation. He organized the Nations Tournament in Buenos Aires in 1939 and he was a permanent representative of the international chess federation in Argentina and represented the country at the Congresses of Paris, London, Warsaw, Stockholm and Buenos Aires. Grau was one of the original signatories in the formation of FIDE at Paris 1924; the 1st unofficial Chess Olympiad. Grau served as Alekhine's second in the match with Capablanca in Buenos Aires 1927.
He was the author of the legendary Tratado General de Ajedrez (General Treatise of Chess) in four volumes that was first published in 1940 and has been translated into numerous languages. The book has been called the Bible of Latin American Chess, and it is highly regarded among Latin American chess players. It is composed of 4 volumes, is now out of print and was never translated to any other language. The 4 volumes cover Fundamentals, Tactics, Pawn Structures and Advanced Strategy.