Eventually, unfortunately, human organs will be grown in pigs. It would be preferable to grow new needed organs in the persons who need the new organs, and I think that’s where we are headed, after the pig phase. Anyway, progess has been made on the pig front:

Part of the difficulty involved timing: Pig embryos develop faster than humans do.

“It’s as if the human cells were entering a freeway going faster than the normal freeway,” said lead investigator Juan Carlos Izpisua Belmonte. “If you have different speeds, you will have accidents.”

After four years of work by more than 40 people, the researchers achieved their goal. Human cells acclimated to pig embryos and grew inside them alongside the pigs’ own parts. 

Link