People donating sperm and eggs will no longer have the right to remain anonymous, under a new law which came into force on Friday.
The law removing anonymity from donors will not be retrospective
Children conceived in this way will now be able to identify their genetic parents once they reach 18.
The new rules will not be retrospective, so people who have already donated will not be affected.
But some experts are concerned that the removal of anonymity will deter donors from coming forward in the future.
And the British Fertility Society has warned that couples who do want eggs or sperm from anonymous donors may choose to go to unlicensed "backstreet" clinics, or travel abroad to countries with less strict regulations.
'No financial obligation'
Around one in seven couples in the UK have fertility problems.
An estimated 7,000 patients receive treatment with donated eggs and sperm, known as gametes, every year and, as a result, 2,000 children are born.
Around 500 sperm donors and 1,500 egg donors are needed each year.
However, clinics say there are long waits for some infertile couples. One reported that a couple had been waiting five years for donated material.
Other clinics have closed their waiting lists.
The change in the rules means that children conceived using donor eggs or sperm will be able to trace their biological parent in the same way as children who are adopted.
While children will be able to access more information about the donor's genetic origins, they will have no financial or legal claim.
Because the law only applies to people who donate from Friday, the first time children born in this way will have the option to ask for the identity of their donor will be when they turn 18 in 2023.
They will have to ask the Human Fertilisation and Embryology Authority to release the information.
The donor will not be able to trace a child.
'Creating a family'
Health minister Stephen Ladyman said: "We think it is right that donor conceived people should be able to have information should they want it about their genetic origins and that is why we have changed the law on donor anonymity."
To limit the possibility of a fall in the number of donors, the Department of Health and the National Gamete Donation Trust (NGDT) has been running a campaign to raise awareness of donation.
Laura Witjens, chair of the NGDT, said evidence from other countries, such as Sweden, which had already removed anonymity rights, showed it was no longer young students who donated.
"There is an initial fall. But then the profile changes. Instead of young single men who do not have children, it tends to be older men, who do have children and who see that what they are doing is creating a family, who come forward."
Dr Ruth Curzon, of the Assisted Conception Unit at Kings Hospital in London, said: "We are going to have to change the way in which we recruit donors. Sperm donation has been seen as 'smutty'. It's time we changed that."
But Dr Allan Pacey, of the British Fertility Society, warned: "There is now serious concern in many clinics about the future of infertility treatments using donated gametes.
"We have evidence that more and more patients are being denied treatment because of a shortage of donor gametes."