Bats began his professional career with FC Sochaux-Montbéliard in 1974, moving on to newly promoted AJ Auxerre the following year. He became undisputed number one at Auxerre (having had to contest the jersey with Albert Rust at Sochaux) and soon found his way into the France squad, being selected for the 1984 European Championship, which France won. By now recognised as a safe, authoritative presence in goal, Bats was also the French number 1 at the 1986 FIFA World Cup, famously saving a penalty from Zico in the quarter-final against Brazil. However, he uncharacteristically let a free-kick from Andreas Brehme slip through his hands in the semi-final against West Germany, which France lost 2–0. He won his last international cap in 1989. At club level, Bats moved on to Paris Saint-Germain F.C. and was their keeper during their first Ligue 1 title in 1986. Bats remained the club's keeper until announcing his retirement in 1992. He then joined the PSG coaching staff as an assistant to Artur Jorge and Luis Fernández before being named trainer in 1997, succeeding Ricardo Gomes. A disappointing 1997–98 season and a series of boardroom changes saw him replaced by Alain Giresse. Following a brief, undistinguished spell in charge of L.B. Châteauroux, he was appointed goalkeeping coach at Olympique Lyonnais in 2000, a post that he still occupies. Bats was stricken with testicular cancer in 1982, but made a full recovery after surgery. As therapy during his convalescence, he took up writing poetry, and has had two volumes of it published.