Born in Ufa and raised in Moscow, Kvyat began competitive kart racing aged 10, before moving to Italy in 2007 to pursue a professional career.

Kvyat was promoted to parent team Red Bull in 2015, taking his maiden podium at the Hungarian Grand Prix and out-scoring teammate Daniel Ricciardo throughout the season.

Despite retaining his seat at Red Bull for 2016 and achieving another podium finish in China, Kvyat was replaced by Max Verstappen following a controversial collision with Sebastian Vettel at the Russian Grand Prix.

He returned to Toro Rosso—later renamed to AlphaTauri—in 2019 to partner Alex Albon and Pierre Gasly, taking his third career podium at the rain-affected German Grand Prix.

After serving as a reserve driver for Alpine in 2021, Kvyat was set to compete in the 2022 FIA World Endurance Championship for G-Drive Racing prior to their withdrawal following the Russian invasion of Ukraine.

[2][3] Rules introduced for the 2014 Formula One World Championship allowed the drivers to pick their own racing numbers that they will carry with them for the rest of their careers.

[5] He also likes heavy metal music, and has cited Metallica's "Whiskey in the Jar" cover and Motörhead's "Ace of Spades" as his two favourite songs.

During the 2005 and 2006 seasons, Kvyat participated in local events, including the Russian karting championship and cup as well as in the occasional races in Italy.

The young driver's parents stayed with him alternately, and he started education in Italian school, continuing to race in local tournaments.

Before the start of the 2008 season, Kvyat, who had already proved himself in the Italian karting scene, signed a contract with the Zanardi factory team, led by well-known manager Dino Chiesa.

In their first race, the new partnership produced an excellent result: Kvyat, who had tested the new kart of the new team just a couple of days before the start of the championship, not only made it to the finals but was fighting for the win.

The Morsicani Racing driver won the prestigious Winter Cup in the KF3 category and then added more victories to his name in the WSK rounds in Sarno and Le-Castellet.

His victory in the Winter Cup was especially outstanding because he made it to the top of the podium after starting from the last position on the grid, where Kvyat found himself as a result of a clutch failure in qualifying.

Kvyat continued to race successfully in karting, taking the bronze at the 2009 European Championship one more time but was unable to reach the finals of the World Cup in Sarno.

Improving his situation during the qualifying heats was barely possible, primarily because the team had decided to place their bets on their second driver, Italian Guliano Maria Niceta.

It was the last win of his career in karting as the Red Bull Junior Team bosses decided to progress Kvyat to the next level, straight into the cockpit of the Formula BMW car for the 2010 season.

Koiranen GP made a wrong call in the mixed conditions and selected wet tyres for Kvyat although the circuit was drying up after the rain; most of the other drivers on the grid started on slick tires.

Kvyat scored his first podium in the series at Hungaroring in July, then won the race in Spa in August to decrease the points deficit between him and the leaders in the drivers' standings.

"[24] Kvyat spent some time adapting to the team and the new car, and returned to good shape, posting several strong performances in qualifying and races.

On 26 September 2017, Toro Rosso announced the decision to replace Kvyat for the forthcoming Malaysian Grand Prix with Frenchman Pierre Gasly, following a sustained run of underwhelming form from the Russian.

"[26] Kvyat returned to racing for Toro Rosso at the United States Grand Prix, following teammate Carlos Sainz Jr.'s move to Renault, and Gasly attending the 2017 Super Formula Championship finale at Suzuka.

[27] In the week between the United States and Mexican Grands Prix, it was confirmed by Helmut Marko that Kvyat would not return to the team and would be released out of the Red Bull driver development programme.

[29] While mainly focusing on the team's simulator at Maranello, Kvyat drove the Ferrari SF71H for the first time at Fiorano, during a Pirelli wet weather test in April 2018.

[33] During the summer break prior to the Belgian Grand Prix, Kvyat's teammate Albon was promoted to Red Bull in August to replace Pierre Gasly, who was demoted back to Toro Rosso.

Red Bull justified the move by saying they wished to trial Albon for the remainder of the season to decide who would partner Max Verstappen at the team in 2020.

[37] Kvyat finished fourth in the 2020 Emilia Romagna Grand Prix after a succession of overtakes on Alexander Albon, Sergio Pérez and Charles Leclerc after the Safety car restart.

[40] Kvyat was due to compete for G-Drive Racing in the LMP2 class in the 2022 FIA World Endurance Championship alongside James Allen and Rene Binder.

[41] However, G-Drive withdrew from the series on 6 March, two weeks before the first race of the season, in response to sanctions imposed by the FIA following the Russian invasion of Ukraine.

He made his NASCAR debut in the 2022 Verizon 200 at the Brickyard at the Indianapolis Motor Speedway road course, finishing in 36th position after retiring from the race with suspension issues.

[55] In 2025, Kvyat was chosen by DS Penske Formula E Team to take part in the Jeddah ePrix rookie free practice session.