He played in 613 NHL games with the Edmonton Oilers, Pittsburgh Penguins, Columbus Blue Jackets, and Mighty Ducks of Anaheim.

On June 23, 2000, Wright was selected in the 2000 NHL Expansion Draft by the newly formed Columbus Blue Jackets.

Since 1987–88, there have been 94 hat tricks scored by a goal (i) at even strength, (ii) on the power play, and (iii) on the penalty kill, by a total of 75 different players.

After four seasons in Columbus, Wright was dealt to the Mighty Ducks of Anaheim on November 15, 2005, in a deal that saw Sergei Fedorov join the Blue Jackets.

In July 2007, Wright rejoined the Columbus Blue Jackets organization in a front office position when he was named a development coach for the club.

On July 11, 2019, it was announced that Wright would depart the Detroit Red Wings and join new GM Ken Holland in Edmonton.