Two couples with a long family history of breast cancer hope to be the first to screen embryos to prevent their future children developing the disease.
Unaffected embryo can be implanted into the mother's womb
What is preimplantation genetic diagnosis?
Preimplantation genetic diagnosis, or PGD, was first introduced in 1990 as an experimental procedure.
It involves removing a cell from an embryo at the eight-cell stage of development, when it is around three-days old, and testing it for genetic disorders so that an unaffected embryo can be implanted into the mother's womb.
What were the rules on PGD?
PGD has traditionally been used to check for genes which will guarantee the resulting child will develop a condition - such as cystic fibrosis and Huntington's disease.
The HFEA has also allowed PGD can be used to check for an inherited form of bowel cancer called familial adenomatous polyposis and retinoblastoma - a very rare form of eye cancer.
It also gave the go-ahead for the technique to be used so that embryos could be selected purely because they are a tissue match for a sick sibling.
Last May, the watchdog ruled it acceptable for doctors to screen embryos for genes such as the breast cancer gene BRCA1, which raise the risk of disease by 60-80%.
Having these genes does not mean someone will definitely develop the conditions - it only means a person will be susceptible to them.
And, unlike the conditions for which PGD is now used, people would not be affected until they were in their 30s or 40s.
The HFEA says any applications to do testing will be assessed on a "case by case" basis.
What are the concerns about PGD?
There are concerns that medicine is moving away from treating diseases and focusing on eliminating the carrier.
Some argue that it might be possible to cure these genetic diseases in the future.
The extension to screening for "susceptibility" genes has also caused concern from disability campaigners, who fear pre-natal selection.