Rigg was born in Doncaster, then in the West Riding of Yorkshire, now in South Yorkshire to Louis Rigg (1903–1968) and Beryl Hilda Helliwell (1908–1981); her father was a railway engineer who had been born in Yorkshire. Between the ages of two months and eight years Rigg lived in Bikaner, India, where her father was employed as a railway executive. Rigg speaks fluent Hindi. She was then sent to a boarding school, the Moravian School in Fulneck, near Pudsey. She disliked her boarding school, where she felt like a fish out of water, but she believes that Yorkshire played a greater part in shaping her character than India did. She trained as an actress at the Royal Academy of Dramatic Art.