Bauer starred in college for the UCLA Bruins, winning the Golden Spikes Award in 2011. That year, he was the third overall selection of the 2011 Major League Baseball draft by the Diamondbacks, and considered a top pitching prospect before making his MLB debut in 2012. The Diamondbacks traded him to the Indians during the 2012–13 offseason.
Bauer attended Hart High School in Santa Clarita, California, and the University of California, Los Angeles (UCLA), where he played for the UCLA Bruins baseball team.
In his freshman year at UCLA, Bauer recorded a 9–3 record with a 2.99 ERA, collecting 92 strikeouts in 105.1 innings.
Bauer was a member of the USA 2009 Baseball Collegiate National Team. He was 1–1 with a 4.67 ERA in five games (three starts), 24 strikeouts, and seven walks in 17 1⁄3 innings. In 2009, he was named to the freshman All-America team by Baseball America.
During the 2010 season, the Bruins had the best record (51–17) in school history and were the second best team in the country. The Bruins played in the 2010 College World Series and were defeated by South Carolina in the NCAA Championship Series. Bauer went 12–3 with a 3.02 ERA, collecting 165 strikeouts in 131 1⁄3 innings. The 165 strikeouts were the best in the nation. In 2010, he was a finalist for the Golden Spikes Award, presented annually to the top amateur baseball player in the country by USA Baseball.
In 2011, Bauer was named the Pac-12 Pitcher of the Year and to the All-Pac-12 First Team. He was also the Collegiate Baseball Newspaper's National Player of the Year. He was named the District IX Player of the Year by the National Collegiate Baseball Writers Association (NCBWA) and College Player of the Year by Baseball America.
In his junior year, Bauer went 13–2 with a 1.25 ERA in 16 starts. He set a nation-leading and Pac-12 single-season record of 203 strikeouts. He finished the season with nine consecutive complete games and established new records at UCLA, including 460 career strikeouts, 34 wins, and 373 1⁄3 pitched innings. Bauer was the recipient of the Golden Spikes Award, and the National Pitcher of the Year Award.
Bauer was selected third overall in the 2011 Major League Baseball Draft by the Arizona Diamondbacks. On July 25, he signed a major league contract with the Diamondbacks, being added to the 40-man roster. He made his professional debut with the Class-A Advanced Visalia Rawhide. In the game, Bauer pitched two innings, allowing one hit, striking out three batters and walked one. He made three starts in total in Visalia, pitching in nine innings and allowing three runs on seven hits. However, he struck out 17 of the 39 batters he faced, earning himself a promotion to the Double-A Mobile BayBears on August 13.
In four starts at AA Mobile, Bauer pitched 16 2⁄3 innings, striking out 26, but walked eight batters and had a 7.56 ERA. He received his first win as a professional August 20, 2011 in a 13–6 victory over the Jacksonville Suns. He was named to appear in the 2012 All-Star Futures Game.
Bauer made his major league debut for the Diamondbacks on June 28, 2012, against the Atlanta Braves. He went 4 innings, struck out 3 batters, and gave up 5 hits in a no-decision. He got his first major league win on July 8, against the Los Angeles Dodgers.
On July 18, 2012, Bauer was optioned back to Triple-A Reno after posting a 1–2 record and 6.06 ERA. The organization rested Bauer for two weeks to keep his arm lively, and he responded by posting six shutout innings in his first start back on August 6.
On December 11, 2012 he was traded to the Cleveland Indians in a three team deal. Bauer went to Cleveland with Matt Albers and Bryan Shaw in exchange for Didi Gregorius, Tony Sipp and Lars Anderson. The Indians also acquired Drew Stubbs in a deal that sent Shin-Soo Choo and Jason Donald to Cincinnati to complete the deal. On June 16, 2015, Bauer got his first hit as a batter against the Chicago Cubs' pitcher Jake Arrieta in the top of the 5th inning at Wrigley Field. During the July 3, 2015, game against the Pittsburgh Pirates' pitcher Antonio Bastardo, Bauer imitated his teammates batting stances (Jason Kipnis, Mike Avilés, and Ryan Raburn) in the top of the 7th inning and drew a walk.
Bauer said he has patterned his overhand delivery after his role model two-time Cy Young Award winner Tim Lincecum. He said he has studied so much video of Lincecum that he can by memory play back frame by frame Lincecum striking out 18 UCLA batters while he was a junior at the University of Washington. "He's been a huge influence on my career and my delivery," Bauer said.
Bauer features an eclectic and large variety of pitches. According to an article in Sports Illustrated, he has experimented with up to 19 different pitches. He explained in an interview why he features so many pitches:
I'm very passionate about my craft and I've always been into science and discovery and all that stuff, so I'm always trying to find a way to get better from season to season. And throwing more pitches is a way that I've found to.... The more pitches that I have, that have different speeds and move differently, the more confusion it creates for the hitter. And if I throw all of them out of the same tunnel and make them look the same though 20 feet of flight ... obviously, I'm going to be a lot tougher to hit.
According to Bauer, he throws the following pitches:Four-seam fastball – thrown at 93–94 mph (tops out at 98)
Changeups – thrown 80–84 mph. Bauer says, "I have two variations of it; I can make it cut or I can make it run."
Curveballs – "curve one" thrown at 74–78 mph, "curve two" thrown at 80–81. According to Bauer, "I have two different grips, one that I use for a strike pitch and then another grip I use when I really want to bury it."
"Dot slider" – a traditional slider, 84–86 mph
"Circle slider" – a slider with movement more similar to a cutter. Bauer: "I use that one primarily early in the count to hit tunnels to righties, disguise it, make it look like a fastball or a changeup and keep it in the zone."
"Reverse slider" – thrown at 88–91 mph, it is designed to act as a "left-handed cut fastball ... It's a cross between a sinking fastball and a screwball—it's a little bit slower than a sinking fastball and a little bit harder than a traditional screwball would be."
Split-finger fastball – a traditional splitter, 86–88 mph
Bauer has gained notoriety for his unusual workout and warmup regimen, which includes long toss at distances of up to 400 feet. Bauer is also known to study his pitching mechanics using high-speed cameras. He has posted a series of videos on YouTube showing his pitching mechanics and repertoire in slow motion.