Adam Richard Fleming was born on 15 May 1948, the son of Major Richard Evelyn Fleming (1910–1977) and the Hon. Dorothy Charmian Hermon-Hodge, daughter of Roland Herman Hermon-Hodge, 2nd Baron Wyfold. He has seven brothers and sisters.
His grandfather was Major Valentine Fleming (1882–1917), and his great-grandfather was the Scottish banker Robert Fleming, founder of the merchant bank Robert Fleming & Co. His uncle on his father's side was the James Bond novelist Ian Fleming.
He was educated at Abberley Hall School and Eton.
He started his career in 1970 as a stock market analyst at Robert Fleming & Co., the company founded by his great-grandfather, earning £7 per week.
In April 2015, The Sunday Times Rich List estimated his family's net worth at GBP £1.5 billion.
Fleming is chairman of Wits Gold, chairman of the Johannesburg Land Company, a director of Zambeef Products, chairman of Fleming Family & Partners, and has farming interests in Zambia and South Africa.
Fleming describes himself as "just a long-term investor with firm beliefs in gold".
He is married to Caroline Wake. They live on a farm in the Cotswolds and have four children, Hector, Nell, Angus and Dickon. His son Hector Fleming is a former director of Standard Chartered's private equity team, co-founded the investment firm Fleming Wulfsohn Africa, and is a director of Imara, The Johannesburg Land Company, Clover Alloys, GoldMoney Networks and Netagio Holdings.
He is a member of White's, London, and the Rand Club of Johannesburg.