After starring as a pitcher for three seasons at William S. Hart High School, Bauer graduated a year early and enrolled at the University of California, Los Angeles (UCLA).

He and fellow ace Gerrit Cole helped lead the UCLA Bruins to a 22-game win streak and a College World Series appearance as sophomores in 2010 where they lost to the University of South Carolina in the CWS finals.

[7] Bauer chose to graduate after his junior season, in part because he meshed poorly with many of his teammates, including future Major League Baseball (MLB) player Mike Montgomery.

Ultimately, however, Bauer failed to displace Josh Collmenter in Arizona's starting rotation, and was assigned back to Double-A Mobile shortly before Opening Day.

[29] Although the Diamondbacks defeated the Atlanta Braves 3–2, Bauer earned a no-decision in his major league debut, pitching four innings before suffering a groin strain that required him to leave the mound.

[31] After four major league starts, during which Bauer went 1–2 with a 6.06 ERA, he was sent back down to Reno, where he was shut down for a period of two weeks in order to preserve his health during his first full season of professional baseball.

Bauer, fellow Diamondbacks pitchers Matt Albers and Bryan Shaw, and Reds outfielder Drew Stubbs went to Cleveland, who in turn sent Shin-Soo Choo and Jason Donald to Cincinnati, and Tony Sipp and Lars Anderson to Arizona.

[37] Bauer began retooling his pitching delivery after the 2012 season, and entered spring training with the Indians with the intention of "overwrit[ing] eight to 10 years of neuromuscular programming".

[44] On May 20, Bauer received his second call-up of the season, replacing Danny Salazar for a start against Detroit Tigers ace and Cy Young Award winner Justin Verlander.

[41] Bauer was named to the Indians' opening day roster for the 2015 season, joining Corey Kluber, Carlos Carrasco, Zach McAllister, and TJ House in the starting rotation.

[49] On July 28, he pitched his first major league complete game, a 2–1 loss against the Kansas City Royals that pushed Cleveland to an eight-game losing streak at home, their longest in 40 years.

[50] As the season progressed, Bauer continued to struggle with allowing home runs, and he was moved to the bullpen on September 17, with Josh Tomlin and Cody Anderson filling his position in the starting rotation.

[41] Despite a strong spring training run, including six scoreless innings in his final Cactus League start, the Indians assigned Bauer to the bullpen to begin the 2016 season, with Anderson and Tomlin remaining in the rotation.

[41] As the Indians advanced through the playoffs, Bauer was scheduled to start Game 2 of the 2016 American League Championship Series (ALCS) against the Toronto Blue Jays, but he suffered an injury to the pinkie finger of his right hand while attempting to repair a homemade drone.

[71] Bauer's push for the AL Cy Young Award was put on pause when he took a line drive to the leg from José Abreu on August 11 during a game against the Chicago White Sox.

[72] Bauer began pitching again on September 22, throwing 1+1⁄3 scoreless innings of relief against the Boston Red Sox so that Cleveland management could decide whether he would be able to start in a potential postseason run.

After recovering from the injuries, he fell ill.[76] On July 28, after allowing the go-ahead run in a game against the Kansas City Royals, Bauer became frustrated and threw a ball over the center field fence of Kauffman Stadium.

Cleveland acquired Yasiel Puig and Scott Moss from Cincinnati and Franmil Reyes, Logan Allen, and Victor Nova from San Diego, while the Reds sent prospect Taylor Trammell to the Padres.

[82] As the season was indefinitely paused, first due to the COVID-19 pandemic and then as negotiations between the league and players' union reached a stalemate, Bauer became frustrated with the lack of play, saying that the suspension had done "irreparable damage" to the sport of baseball.

[84] As the regular season built to a close, Bauer began pitching on short rest, starting every fourth day rather than every fifth to help the Reds clinch a postseason berth.

[86] Although Atlanta swept Cincinnati during the Wild Card series, Bauer bested Yu Darvish of the Cubs and Jacob deGrom of the New York Mets in voting for the NL Cy Young Award.

[88] After becoming a free agent, Bauer courted offers from a number of MLB teams, ultimately turning down a deal from the New York Mets to sign with the Los Angeles Dodgers on February 5, 2021.

[97] On April 29, 2022, Rob Manfred, the Commissioner of Baseball, announced that Bauer would be suspended for 324 games, equivalent to two seasons, without pay for a violation of the league's Joint Domestic Violence, Sexual Assault and Child Abuse Policy.

[103] On March 13, 2023, Bauer agreed to an incentive-laden one-year, $4 million contract with the Yokohama DeNA BayStars, a Central League team in Nippon Professional Baseball.

[135] Speculation that Bauer had been using grip-enhancing substances increased after the spin rate on his fastball dropped by more than 200 rpm following an announcement from MLB that the league would begin enforcing their rules on pitch doctoring.

[139] Momentum is also a YouTube group by Bauer featuring content creators such as former San Francisco Giants minor leaguer Eric Sim, Kevin Chan, Cole Acheronti, Tosh Semlacher, and himself in videos, alongside his agent Rachel Luba behind the scenes.

[145] In February 2017, Bauer accused Apple and Twitter of exhibiting a liberal media bias, after which he proceeded to express a variety of political beliefs, including climate change denial, skepticism that Obama was born in the United States, and a defense of the Indians' mascot Chief Wahoo, which has been criticized as a racial caricature.

[150] In June 2021, Bauer was placed on administrative leave and investigated by the Pasadena Police Department for the alleged assault of a San Diego woman earlier that year.

[152] He remained under investigation by MLB and police through the 2021 season[153] but the Los Angeles County District Attorney's Office elected not to file criminal charges, citing a lack of evidence.

[157][158] While Bauer was under investigation in 2021, The Washington Post uncovered court records showing that a separate woman from Ohio had sought a temporary order of protection in June 2020.