As a teenager, Connolly showed off his promise with his hometown team, Prince George in the Western Hockey League. At 16, he scored 30 goals and 60 points during his first full season in juniors to become the first WHL player since Patrick Marleau in 1995-96 to reach 30 goals at that age. Connolly was named the rookie of the year in the WHL and the Canadian Hockey League.

Connolly is a big, skilled right wing with good offensive instincts who can play up and down the lineup. He showed his versatility during the 2018 Stanley Cup Playoffs, scoring nine points (six goals, three assists) in 24 games to help the Washington Capitals win their first Stanley Cup championship since entering the NHL in 1974. Two of the goals came during the Final against the Vegas Golden Knights and he had the primary assist on Lars Eller's Cup-winning goal with 7:37 left in the third period of Game 5 at T-Mobile Arena.

Connolly is a big, skilled right wing with good offensive instincts who can play up and down the lineup. He showed his versatility during the 2018 Stanley Cup Playoffs, scoring nine points (six goals, three assists) in 24 games to help the Washington Capitals win their first Stanley Cup championship since entering the NHL in 1974. Two of the goals came during the Final against the Vegas Golden Knights and he had the primary assist on Lars Eller's Cup-winning goal with 7:37 left in the third period of Game 5 at T-Mobile Arena.

As a teenager, Connolly showed off his promise with his hometown team, Prince George in the Western Hockey League. At 16, he scored 30 goals and 60 points during his first full season in juniors to become the first WHL player since Patrick Marleau in 1995-96 to reach 30 goals at that age. Connolly was named the rookie of the year in the WHL and the Canadian Hockey League.

A hip injury in his draft year limited Connolly to 16 games, but that didn't stop the Tampa Bay Lightning from taking him with the No. 6 pick in the 2010 NHL Draft.

He returned to Prince George for the 2010-11 season and was captain, scoring 46 goals and 73 points in 59 games. Connolly made the Lightning roster at training camp 2011 but saw limited ice time and scored 15 points (four goals, 11 assists) in 68 games.

Connolly played most of the next two seasons in the American Hockey League. In 2012-13 he scored 63 points (31 goals, 32 assists) and helped Syracuse reach the Calder Cup Final with six goals and 11 points in 18 playoff games. He also was named to the AHL Second All-Star Team.

Connolly made the Lightning's opening-night roster for the second time in 2014-15, but primarily played on the third line and averaged less than 12 minutes of ice time per game before he was traded to the Boston Bruins on March 2, 2015.

He found a role as third line forward with the Bruins in 2015-16, scoring 25 points (nine goals, 16 assists) in 71 games. Connolly signed with the Capitals as a free agent July 1, 2016 and scored 15 goals in each of the next two seasons, and 22 in 2018-19, before signing with the Florida Panthers on July 1, 2019.

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