At 6-foot-2 and 193 pounds, Connolly is a big, skilled right wing with strong offensive instincts who can play up and down the lineup.

As a teenager, Connolly showed that promise with his hometown team, Prince George, in the Western Hockey League. At 16 he scored 30 goals and 60 points in his first full season in juniors, becoming 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 Canadian Hockey League (which comprises the three major junior leagues in Canada).

A hip injury in his draft year limited Connolly to 16 games, but that did not hurt his ranking as one of the top prospects available at the 2010 NHL Draft, where the Tampa Bay Lightning selected Connolly with the No. 6 pick.

During his time with the Lightning, Connolly had his ups and downs.

He returned to Prince George for the 2010-11 season and was captain, scoring 46 goals and 73 points in 59 games. Connolly would then make the Lightning roster out of training camp in 2011, but saw his ice time limited as he became a frequent healthy scratch. He eventually was loaned to Team Canada for the World Junior Championship in December and January.

Connolly was among the last cuts from training camp for each of the next two seasons, starting out each in the American Hockey League. In 2012-13 he had 31 goals and 63 points in 71 games and helped Syracuse reach the Calder Cup Finals with six goals and 11 points in 18 playoff games. He also was named to the AHL Second All-Star Team.

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

Before cracking the Bruins lineup, Connolly broke his hand during practice and missed all but the final five games of the season. After playing 71 games in 2015-16 with Boston, Connolly signed a one-year, free agent contract with the Washington Capitals.