Curran was born in Northampton, the youngest son of former Zimbabwe international cricketer Kevin Curran. He is the brother of Surrey and England cricketer Tom Curran, and Northamptonshire cricketer Ben Curran. He was educated at Springvale House, St. George's College, Harare, and Wellington College, Berkshire. He lived on the family farm in Rusape, before it was seized under the land reform policy in 2004.

Curran represented Surrey at Under-15, Under-17, and Second XI level. During the 2014 season he represented Weybridge in the Surrey Championship Premier Division. He was described by Surrey director of cricket Alec Stewart as "the best 17-year-old cricketer that I have seen".

Curran made his senior debut in a Twenty20 match in the NatWest t20 Blast tournament against Kent at The Oval on 19 June 2015, aged 17 years and 16 days. He made his first-class debut in a County Championship fixture against Kent at The Oval on 13 July 2015. At the age of 17 years and 40 days he became Surrey's second youngest first-class cricketer in history after Tony Lock, who debuted exactly 69 years earlier at the age of 17 years and 8 days, also against Kent at The Oval. He returned figures of 5/101 in the first innings, and is believed to be the youngest-ever player to take five wickets in a County Championship match. He made his List A debut in a Royal London One-Day Cup match against Northamptonshire at The Oval on Thursday 27 July 2015.

He was signed by Auckland Aces for the 2017–18 Super Smash.

Curran represented Zimbabwe Under-13s at the 2011–12 CSA Under-13 Week tournament.

He represented England Under-19s at the 2016 ICC Under-19 Cricket World Cup, where he played all six games, scoring 201 runs and taking seven wickets to help his team finish sixth. He was selected for the England Lions for their 2016–17 tour of the United Arab Emirates, and again for their match against South Africa A at Canterbury in the 2017 season.

Curran received his first senior call-up for England in January 2018 for the 2017-18 Trans-Tasman Tri-Series against Australia and New Zealand,^[11] but did not play any games.

On 30 May 2018 he was added to the England Test squad ahead of the second Test against Pakistan, as cover for Ben Stokes.^[12] He made his Test debut at Headingley on 1 June 2018.^[13] Curran scored 20 in England's only innings, and returned match figures of 2/43..^[14]

On 24 June 2018, he made his One Day International debut against Australia.^[15]

Curran retained his place in the England squad for the Test series against India. In the first Test at Edgbaston he took 4/74 in the first innings, including the wickets of India's top three batsmen^[16], and scored 63 in England's second innings, and was awarded Player of the Match.^[17] After being omitted from the England team for the third Test, he returned in place of the injured Chris Woakes at the Ageas Bowl, where he top scored in England's first innings with 78.^[18] Curran registered his first Test duck in the fifth Test at The Oval, but was named England's Player of the Series against India, having contributed 272 runs and 11 wickets in England's 4-1 series victory.