Norton was selected in the 1984 NHL Entry Draft by the New York Islanders.

Norton played for Team USA Hockey in the 1988 Winter Olympics in Calgary, Alberta.

Norton has also played for the San Jose Sharks (three separate times), St. Louis Blues, Edmonton Oilers, Tampa Bay Lightning, Florida Panthers (two separate times), Pittsburgh Penguins, and Boston Bruins.

[1] Norton and Rich Pilon are depicted as the two defenders Mario Lemieux skates between in a 4,700-pound bronze statue unveiled on March 7, 2012, at PPG Paints Arena (then called Consol Energy Center) in Pittsburgh, Pennsylvania.

[2] A graduate of the University of Michigan, he grew up in the town of Acton, Massachusetts and attended Cushing Academy.