Davidson officiated in the 1992 World Series, as well as the National League Championship Series in 1988, 1991 and 1996. He also worked in the National League Division Series in 1995, 1998 and 2009, and in the 1987, 1993, and 2014 All-Star Games. Nicknamed "Balkin' Bob" or "Balk-a-day-Bob" due to his frequent balk calls, Davidson was voted by players and managers as one of the worst umpires in MLB, placing fourth-worst in a "Sports Illustrated" poll in 2011.