Travis Shaw was born and raised in the city of Washington Court House, Ohio, and attended Washington High School. He played college baseball at Kent State University in Kent, Ohio. He batted a .307 average with a Mid-American Conference-leading 14 home runs while playing 62 games for the Golden Flashes in 2011. He also slugged .553 and collected 51 runs batted in, being named to the first team All-MAC in 2011 and the second team in 2010.
Shaw also played in the Cape Cod Baseball League for the Bourne Braves. The Red Sox selected Shaw in the ninth round of the 2011 MLB draft. On July 7, 2015, Shaw was called up from Triple A Pawtucket and recorded his first major league hit. Shaw eventually went 3 for 4 on the night with a run scored. On December 6, 2016, the Red Sox traded Shaw to the Milwaukee Brewers.
On April 3, 2017, Shaw was the starting third basemen, making his Brewers debut on Opening Day against the Colorado Rockies. He went 2-4, hitting two doubles, including an RBI double. His father, Jeff, is a former two-time All-Star pitcher who played for six major league teams.