Children may inherit mitochondrial DNA defects from their mothers, but not their fathers.
People with faulty mt DNA can develop symptoms or be carriers of the condition without experiencing ill effects themselves.
Pro-Nuclear Transfer (PNT) is similar but in this case the mother's egg is fertilised first.
Its nuclear DNA is then transferred to a fertilised donor egg, containing healthy mitochondria, whose own nuclear DNA has been removed. There is evidence that small numbers of faulty mitochondria can be 'carried over' into the child.
They might even replicate in the developing embryo.
However, this is not thought to pose a serious risk.
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Critics of the treatment say it is a dangerous step that will lead to the creation of genetically modified "designer babies".
But the medical team in Newcastle said they were delighted with the decision "to help families affected by these devastating diseases".
Dr Zhang carried out the procedure at the Mexican clinic Mitochondria are tiny rod-like structures in cells which act as power houses, generating the energy that allows our bodies to function.