He played for three MLB teams—the Boston Red Sox, Milwaukee Brewers and Toronto Blue Jays—and was nicknamed the Mayor of Ding Dong City.

Shaw attended Washington High School, the alma mater of his father, Jeff, a former MLB closer who was a two-time All-Star who played for six major league teams during an 11-year career.

He was promoted to the Double-A Portland Sea Dogs early in August, but was strongly tested in the upper Eastern League, where he finished with a .227 average, three home runs and 12 RBI in 31 games.

Then, during the offseason he had a chance to play against some of the best prospects in baseball in the Arizona Fall League, where he hit a .361/.452/.705 line with five home runs and 19 RBI in 17 games for the Surprise Saguaros team.

He hit .305/.406/.548 with 11 home runs and 37 RBI in 208 plate appearances for Portland[6] before gaining a promotion to the Triple-A Pawtucket Red Sox.

[6] Overall in 2014, Shaw topped the Sox minor league system with 21 home runs and collected 78 RBI, being surpassed only by Carlos Asuaje (101).

On July 7, he was called back up and recorded his first major league hit, a second inning single against the Miami Marlins.

[citation needed] During spring training in 2016, manager John Farrell stated that there would be open competition for the starting third baseman's job between Shaw and the struggling Pablo Sandoval.

On December 6, 2016, the Red Sox traded Shaw, Mauricio Dubón, and Josh Pennington to the Milwaukee Brewers for Tyler Thornburg.

[13] On April 3, 2017, Shaw was the starting third baseman, making his Brewers debut on Opening Day against the Colorado Rockies.

[14] Shaw went on to have a tremendous season with the 2017 Brewers, batting .273, with 31 home runs and 101 RBIs, along with 10 stolen bases in 144 games played.

On February 16, 2021, Shaw signed a minor league contract with the Milwaukee Brewers organization that included an invitation to spring training and an opt-out date of March 15.

[23] On August 15, 2021, Shaw was claimed off waivers by the Red Sox;[24] he was added to the team's active roster the next day.

[25] The following week, his first hit since rejoining the team was a walk-off grand slam to give Boston an extra-innings win over the Texas Rangers.

[23] He also appeared in five postseason games, going 1-for-5 as a pinch hitter, as the Red Sox advanced to the American League Championship Series.