Roberts was selected by the Calgary Flames in the first round (No. 12) of the 1984 NHL Draft after the forward from North York, Ontario, scored 24 goals in 57 games in his second season with Ottawa of the Ontario Hockey League. He played most of his next two seasons in the OHL before turning pro and making his NHL debut with the Flames on Nov. 11, 1986, when he scored a goal against the Vancouver Canucks.

Roberts became a full-time NHL player in 1987-88 and scored 22 goals in 1988-89, when the Flames won the Stanley Cup for the first time. In 1989-90, he scored 39 goals and 72 points, and two seasons later set NHL career highs in goals (53) and points (90).

He scored 38 goals in 1992-92 despite being limited to 58 games by a quadriceps injury, then scored 41 goals in 1993-94 despite missing the final two weeks of the season because of a nerve injury in his neck. That injury limited him to eight games in 1994-95; he had surgery twice in 1995 (March, October) to try to correct the problem.

Roberts returned to action on Jan. 10, 1996 and scored a goal against the Hartford Whalers. He finished with 42 points (22 goals, 20 assists) in 35 games and won the Masterton Trophy for perseverance and dedication to hockey, but issues with his neck remained and Roberts announced his retirement on June 17, 1996.

A physiotherapy regimen enabled Roberts to return for the 1997-98 season. At his request, the Flames traded Roberts to the Carolina Hurricanes on April 25, 1997. He was a two-time 20-goal scorer in three seasons with the Hurricanes and signed with the Toronto Maple Leafs as a free agent on July 4, 2000.

Roberts led Toronto with 29 goals in 2000-01 and followed that with 21 in 2001-02. He scored seven goals and finished with 19 points in 19 playoff games, helping the Maple Leafs reach the Eastern Conference Final before losing to the Hurricanes.

Shoulder surgery limited Roberts to 14 games in 2002-03, but he was healthy again in 2003-04 and scored 28 goals. He was also selected to NHL All-Star Game for the third and final time and played his 1,000th NHL game on Jan. 13, 2004.

Roberts signed with the Florida Panthers on Aug. 1, 2005 and was traded to the Pittsburgh Penguins on Feb. 27, 2007. He helped the Penguins reach the Stanley Cup Final in 2008, which they lost to the Detroit Red Wings in six games.

He joined the Tampa Bay Lightning for the 2008-09 season but sustained an elbow injury and announced his retirement on March 10, 2009. Roberts finished with 910 points (438 goals, 472 assists) in 1,224 NHL games and 93 points (31 goals, 62 assists) in 130 playoff games.

After retiring, Roberts became a well-known personal trainer whose clients included several of NHL players.