Conway was born in Bromley, London, and attended Sevenoaks Prep School from 1986 to 1996. He resides in Sevenoaks, Kent.
Conway began karting at the age of eight at Rye House in Hertfordshire. He progressed to become the Formula A British Karting Champion before moving into Formula Ford with Van Diemen. In 2004, Conway was crowned Formula Renault UK Champion, competing with Fortec Motorsport. He continued with Fortec in the 2005 British F3 International Series, finishing third overall behind Álvaro Parente and Charlie Kimball.
In 2006, Conway joined Räikkönen Robertson Racing. He was signed by 2MB Sports Management run by former Grand Prix drivers Martin Brundle and Mark Blundell. He dominated the British F3 International Series, securing the title with three races remaining. He also finished first among the British F3 drivers in Race Two of the Pau Grand Prix in France, won overall by Romain Grosjean. At the Macau Grand Prix that year, Conway qualified fourth and finished seventh in the qualifying race, but retired from the main race with engine failure, classified fourteenth. He won the National Racing Driver of the Year Award at the 2006 McLaren Autosport Awards.
Conway debuted in GP2 at Silverstone in June 2006, substituting for the injured Olivier Pla in a DPR Direxiv car, finishing eleventh in both races. He signed with Super Nova Racing for the full 2007 GP2 Series season, achieving a podium finish at Silverstone and becoming a test driver with the Honda Racing F1 Team. In 2008, he joined Trident Racing, after testing for several other teams. He won the sprint race at Monaco after starting eighth in the feature race. He finished twelfth in the drivers' championship that year, an improvement of two positions from 2007.
In 2008, Conway tested an IndyCar Series car at Infineon Raceway, impressing several regulars. He signed with Dreyer & Reinbold Racing for a full-time IndyCar Series campaign in 2009.
The 2010 Indianapolis 500 saw Conway involved in a dramatic final-lap crash while battling Ryan Hunter-Reay for the lead. He suffered a broken leg, a compression fracture of a thoracic vertebra, and was fitted with a back brace, ending his season. In 2011, he signed with Andretti Autosport, winning at Long Beach but ultimately finishing seventeenth in the championship after failing to qualify for the Indianapolis 500.
Conway moved to A. J. Foyt Enterprises for the 2012 season. During a pit stop, contact with a crew member damaged his front wing, but went unnoticed initially. Later, he lost control and collided with Will Power, resulting in a spectacular airborne crash. He was uninjured but unable to finish the race. He finished third in Toronto, his best result of the season. Prior to the final race at Fontana, Conway informed Foyt that he was uncomfortable racing on oval tracks, stemming from injuries sustained at the 2010 Indianapolis 500 and discussion surrounding oval racing following Dan Wheldon’s death in 2011. He was replaced by Wade Cunningham.
In 2013, Conway ran a one-off race at Long Beach with Rahal-Letterman-Lanigan Racing, finishing 25th due to an electrical issue, and then signed with Dale Coyne Racing for a doubleheader event, winning at Belle Isle Park. In 2014, he won at both the Toyota Grand Prix of Long Beach and the Honda Indy Toronto Doubleheader while driving for Ed Carpenter Racing, competing only in road and street circuit races.
In 2013, Conway supplemented his IndyCar career by joining G-Drive Racing in the LMP2 class of the FIA World Endurance Championship. He and co-drivers John Martin and Roman Rusinov achieved multiple podiums, but were excluded from a third-place finish at Le Mans due to a fuel tank infraction. They won at São Paulo and Austin, and finished third in the category standings.
He joined Toyota Gazoo Racing as a test and reserve driver in 2014, scoring his first overall win in sportscars at Bahrain. He also briefly joined Dragon Racing for the inaugural Formula E season before leaving the team in September.
Conway became a full-time Toyota driver in 2015, partnering with Stéphane Sarrazin and Alexander Wurz, finishing sixth in the standings. With Wurz’s retirement, he partnered with Sarrazin and Kamui Kobayashi for 2016, finishing second at Le Mans and third in the championship. In 2017, alongside Kobayashi and José María López, he retired from Le Mans due to a clutch failure and finished last of the full-time LMP1 entries.
The 2018–19 season saw Conway, Kobayashi, and López consistently battling the sister Toyota car. They won at Fuji and Shanghai, finishing second in the championship behind the No. 8 Toyota. In 2019–20, they won at Silverstone and Bahrain, securing the WEC title.
In 2021, with the introduction of the Le Mans Hypercar formula, Conway, Kobayashi, and López won at Monza and Le Mans, also claiming the World Championship. The 2022 season brought a win at Spa, but a second-place finish at Le Mans due to an axle motor-generator issue. They won again at Bahrain, finishing third in the overall standings.
The 2023 season saw a victory at Sebring and Fuji, but a retirement at Le Mans. They finished second in the championship. In 2024, Conway missed Le Mans due to a cycling accident, being replaced by López. He returned to win at Bahrain. In 2025, Conway was again sidelined with a collar bone injury, again being replaced by López, and finished sixth at Le Mans. He would return to win at Bahrain.
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