Ubilava won the Georgian Chess Championship in 1974 and 1986 and played for Georgia in the 1992 Chess Olympiad. His other successful performances include 1st at Trencianske Teplice 1985, 1st at Eforie Nord 1988, =1st at Tbilisi 1988, 1st at Benasque 2001, 3rd at Elgoibar 2003, =2nd at La Roda 2005, =1st at Benasque 2005 and =1st at Almeria 2008.