The big risk with transplanting organs from one person to another is that the body may reject it, as the immune system is programmed to attack foreign object, This is the reason why, when possible, family members are the best donors as they have similar DNA.
But because not every organ can be donated from a living person (not without killing them) this is not always possible. And even when in case with organs that can be donated, family may not be willing to donate as it may put them at risk.
To add to this is the fact there is a shortage of organ donors. Those in need of organs are put on waiting list with hopes an organ becoming available.
The possible solution to this could be the cloning of human organs