The Edinburgh University-led research has been described as the largest genetic study of hair colour to date.
It had been thought red hair was controlled by a single gene, MC1R, with versions passed on from both parents.
However, not everyone carrying two red-haired versions is a redhead, meaning that other genes had to be involved.
The team examined DNA from almost 350,000 people who had taken part in the UK Biobank study.
When they compared redheads with people with brown or black hair, scientists identified eight previously-unknown genetic differences that are associated with ginger locks.
The team also looked at the functions of the genes they identified and found some of them work by controlling when MC1R is switched on or off.
The researchers also uncovered differences in almost 200 genes associated with blondes and brunettes.
Prof Albert Tenesa, of the university's Roslin Institute, said: "We are very pleased that this work has unravelled most of the genetic variation contributing to differences in hair colour among people."
Prof Ian Jackson, of the medical research council human genetics unit at the university, said: "We were able to use the power of UK Biobank, a huge and unique genetic study of half a million people in Britain, which allowed us to find these effects."
The study, published in Nature Communications, was funded by the Medical Research Council and the Biotechnology and Biological Sciences Research Council.
More about: Gene