Non-urgent cases may have to wait up to nine years
It follows an admission that the Welsh Centre for Burns and Plastic Surgery at Morriston Hospital, Swansea, had the longest waiting list in the developed world.
The Welsh Health Minister Jane Hutt has asked the assembly's Specialist Health Services Commission to review the situation.
Hamish Laing, the centre's clinical director, said they should have more than 20 surgeons, but only had seven.
As a result, he said less urgent cases had been pushed to the back of the list, which stood at 1,800 people.
The average waiting time for plastic surgery in England was 18 months.
A spokesman for Ms Hutt said: "The Welsh Assembly is aware that some patients have to wait an unacceptable length of time, and there are issues surrounding the provision of plastic surgery at Morriston.
"We have asked for an urgent review of the requirements for services and the way they are delivered."
Andrew Bellamy, director of corporate development at the Swansea NHS Trust, said attempts were being made to rectify the situation.
"It is a cause of great concern to clinicians and managers within the trust that waiting lists for plastic surgery are so long.
"We need a significant increase in the number of plastic and reconstructive surgeons to cover the population served.
"But of course we will require a planned programme of investment to ensure the necessary facilities are available, and that we are able to recruit for the service."
The assembly has set a target waiting time for in-patient treatment of 15 months.
David Melding, Conservative spokesman on health in the assembly, said: "This is a record even for the Welsh NHS."
"These people have some level of disfigurement and it is very disturbing they have to wait six to nine years for non-urgent treatment.
"This is the most acute example, but it does reflect the long waiting lists across the NHS."