Reeve was born in Hong Kong in 1963 to parents who working in the territory as teachers. His mother, Monica, later served as a scorer for a number of England Test matches. He also played for KJS football team alongside captain of the team Steven Markbreiter, who now coaches the Hong Kong based team Sai Kung Bulldogs.

Reeve first played cricket in England as a member of the Marylebone Cricket Club (MCC) Young Cricketers – an academy of up and coming young players between the ages of 18 and 20, who are based at Lord's. He signed for Sussex in June of the 1983 English season, and took 42 wickets in the County Championship at 29.35 apiece. He was given little opportunity with the bat and batted at number 10 but performed the duties of night watchman. This helped Reeve show his batting ability where in this role he made his maiden first class hundred against Surrey at Guildford. He remained with Sussex for six seasons, his most successful being 1987, when he managed a batting average of over 40 and took 42 wickets at under 30.

His lack of a high enough batting position at Sussex led to his move to Warwickshire in 1988, and it was here that he gained his greatest successes. In the batsman-friendly summer of 1990, he averaged 54, making his highest first-class score of 202 not out (against an attack including Curtly Ambrose), and making two other centuries on the way to a total of 1,412 runs. He averaged over 40 for Warwickshire during his time at the Midlands club proving the move a good decision and helping him gain international selection. Reeve was made captain in 1993, and in 1994 led the county to an unprecedented domestic treble, winning the County Championship, the AXA Equity and Law League and the Benson & Hedges Cup, only being defeated in the final of the NatWest Trophy. This achievement came despite well-publicised differences of opinion with Warwickshire's star batsman, Brian Lara, vividly described a few years later by Reeve in his book Winning Ways. He was known for his inventive cricket brain and once held the bat left handed and swept Ravi Shastri against Glamorgan. Today known as the switch hit. One of his improvisations against left-arm spinner, Rajesh Maru, of Hampshire was to drop his bat to avoid being caught out by the close fielders.

The following season, Warwickshire retained their Championship and NatWest Trophy crowns, but Reeve left Warwickshire after playing until halfway through the 1996 season. Reeve made over £400,000 tax free during his benefit year with Warwickshire. Reeve was named as one of the Wisden Cricketers of the Year in 1996, and received the OBE for services to cricket that same year. He moved on to become coach to Somerset (for whom he also played several games in one-day cricket in 1998), and also began to work in television cricket commentary for Channel 4.

Reeve played for Hong Kong at the 1982 ICC Trophy, averaging 34.50 with the bat and 15.71 with the ball. He returned to international cricket in 1991, playing for England. Reeve played only three Test matches for England, but his improvising style was better suited to One Day Internationals and he made 29 appearances in this form of the game, appearing in both the 1992 and 1996 World Cups. However, he never scored highly in ODIs, and his batting average would have been considerably lower, but for his high proportion of not-out innings.

Following his retirement from cricket, Reeve was involved with the Channel 4 broadcasting coverage of cricket from the start when the channel won the rights from the BBC to show England Tests in the UK. Duties included fronting the studio discussions with pundits, commentary, introducing and presenting awards af