He played in Major League Baseball (MLB) for the Oakland Athletics, San Diego Padres, and Tampa Bay Rays.

At Goddard High, Norris played third base before transitioning to catcher, and also won a Class 6A Championship title.

[4] Norris spent the 2007 season with the GCL Nationals, Washington's affiliate in the rookie-level Gulf Coast League.

[6] On December 23, 2011, the Nationals traded Norris, A. J. Cole, Tommy Milone, and Brad Peacock to the Oakland Athletics for Gio González and Robert Gilliam.

He went 0 for 3, but made a key defensive play in the ninth inning throwing out Dodgers Dee Gordon attempting to steal second base.

[9][10] When Suzuki was traded to the Washington Nationals on August 3, Norris became the primary catcher for the team, backed up by the newly acquired George Kottaras.

[12] In 2013, Norris was the primary catcher in a catching platoon, backed up by left-handed hitters John Jaso and Stephen Vogt.

[15] He hit a slash line of .270/.361/.403 with 10 home runs across 385 at-bats, while making 93 starts at catcher with John Jaso as a back-up.

[16] On December 18, 2014, the Athletics traded Norris and Seth Streich to the San Diego Padres in exchange for R. J. Alvarez and Jesse Hahn.

[23] After the team signed free agent catcher Matt Wieters, the Nationals reportedly attempted to trade Norris but were unable to find a taker.

[12] On September 1, 2017, Norris was suspended for the remainder of the 2017 season for violating MLB's personal conduct policy, regarding a domestic violence case against his former fiancée.

[31] On April 14, 2018, Norris signed with the Sugar Land Skeeters of the Atlantic League of Professional Baseball.