Smart is the son of Camellia Smart and Billy Frank Smart. He has three older brothers: Todd Westbrook (deceased), Jeff Westbrook, and Michael Smart. He attended Edward S. Marcus High School in Flower Mound, Texas, along with one of his former Oklahoma State teammates, Phillip Forte. During his senior year, Smart averaged 15.1 points, 9.2 rebounds and 5 assists. In his high school career, he achieved a record of 115–6 through three seasons and was a two-time 5A state champion. He was also named a McDonald's All-American and was an ESPNHS first team All-American. Smart played competitive football until the sixth grade and enjoys playing tennis in his spare time.
Considered a five-star recruit by ESPN.com, Smart was listed as the No. 1 shooting guard and the No. 10 player in the nation in 2012.
During his freshman year at Oklahoma State, Smart led the Cowboys to a 24–8 record and finished third place in the Big 12 behind Kansas and Kansas State. Smart averaged 15.4 points, 5.8 rebounds, and 4.2 assists per game and led the Big 12 in steals, where he recorded 99 and averaged 3.0 per game. Smart and the Cowboys earned a trip to the NCAA bracket that year, clinching a #5 seed in the Midwest Region. During the first round of the tournament though, the Cowboys were eliminated by the #12 seed Oregon. On April 17, 2013, Smart held a press conference in the student union at OSU and announced that he would not declare for the NBA draft and instead, return to OSU for his sophomore season.
On November 19, 2013, Smart tied an OSU single game scoring record with 39 points leading his #7 Oklahoma State Cowboys past #11 Memphis. On February 8, 2014, during a game at Texas Tech, Smart shoved a fan in the stands after a verbal altercation in the closing minutes of the game, and received a technical foul. Reports after the game stated that Smart claimed the fan yelled a racial slur at him. At a press conference the following afternoon, Smart would not comment on that element of the altercation, and coach Travis Ford chose not to address it. It was soon discovered that the fan had not used a racial slur, but even though Smart was deceptive in his original comments, the fan did admit to having spoken inappropriately to Smart by calling him "a piece of crap". Smart was subsequently suspended for three games and the fan agreed not to attend any further Texas Tech games during the 2013–14 season. Five days later, Smart was named one of the 30 finalists for the Naismith College Player of the Year. In the first game of the 2014 NCAA tournament, the Cowboys lost to Gonzaga. He finished with 23 points, 13 rebounds, 7 assists and 6 steals, becoming the first player in tournament history to record 20 points, 10 rebounds, 5 assists and 5 steals.
On April 7, 2014, Smart declared for the NBA draft, forgoing his final two years of college eligibility.
On June 26, 2014, Smart was selected with the sixth overall pick in the 2014 NBA draft by the Boston Celtics. He joined the Celtics for the 2014 NBA Summer League, and signed with the team on July 10. In just his fifth NBA game, Smart sprained his left ankle in the Celtics' 101–98 win over the Indiana Pacers on November 7. He underwent an MRI following the game and was ruled out for two to three weeks. After missing ten games with the injury, Smart returned to action on December 3 against the Detroit Pistons. On December 4, he was assigned to the Maine Red Claws of the NBA Development League. He was recalled the next day after playing in Maine's win over the Erie BayHawks. On March 18, 2015, he scored a season-high 25 points in a loss to the Oklahoma City Thunder. On March 21, he was suspended for one game without pay for hitting San Antonio Spurs forward Matt Bonner in the groin the previous night. On May 18, Smart was named to the NBA's All-Rookie Second Team, garnering 142 points in the voting process.
On July 16, 2015, while playing for the Celtics at the 2015 Las Vegas Summer League, Smart dislocated two fingers on his right hand. On November 15, 2015, he scored a career-high 26 points in a 100–85 win over the Oklahoma City Thunder. Between November 22 and December 26, Smart missed 18 games with a lower left leg injury. He returned to action on December 27 against the New York Knicks, scoring six points in 13 minutes off the bench. On January 15, 2016, in a win over the Phoenix Suns, Smart recorded his first career triple-double with 10 points, 11 assists and 11 rebounds, becoming the first Celtics player to record a triple-double off the bench since Art Williams did so in 1971. On January 31, he tied his career high of 26 points in a loss to the Orlando Magic.
On November 9, 2016, Smart scored a then season-high 20 points in a 118–93 loss to the Washington Wizards. On December 25, he scored 15 points and made a tiebreaking 3-pointer with 47 seconds left to help the Celtics claim a 119–114 win over the New York Knicks. On January 7, 2017, he scored a season-high 22 points in a 117–108 win over the New Orleans Pelicans. In Game 3 of the Eastern Conference Finals on May 21, Smart, who started in place of the injured Isaiah Thomas, made seven 3-pointers and scored 27 points to help the Celtics defeat the Cleveland Cavaliers 111–108. Boston were blown out in the first two games of the series and came back from a 21-point deficit in the third quarter of Game 3. They went on to lose the series in five games.