Bedřich Ščerban’s high-level hockey career lasted from 1983 to 2008 with multiple European teams. He competed at three Winter Olympics (1988, 1992, 1994) and eight World Championships (1987, 1989-95) for Czechoslovakia and the Czech Republic, winning five bronze medals – 1992 Olympic and World bronzes in 1987, 1989, and 1992-93. He was also on the 1988 Czechoslovakian Canada Cup team.
Playing defense, Ščerban played most of his Czech career with HC Dukla Jihlava – parts of 10 seasons – but he also played in his native land for VHK Vsetin, HC Berounští Medvědi, and SK Horácká Slavia Třebíč, before retiring with HC Dukla Jihlava after the 2007-08 season. But he hopped around Europe with various leagues and teams, playing for Tappara Tampere in Finland, Brynas IF Gävle in the Swedish league, and EHC Essen Moskitos, EHC Freiburg and EHC Freiburg in the German Elite League. While playing for HC Dukla Jihlava in 1991 Ščerban won the Golden Stick award, given to the top Czechoslovakian hockey player.