Organs that can be given by living donors include the kidney and portions of the liver, lung, intestine, and pancreas. Kidney transplants are the most common; they have now been done for more than 50 years. The first one was done in 1954 in Boston at what is now Brigham and Women's Hospital.
The care for living organ donors – before after after the transplant – is top-rate. The screening that is done for living donors can, in fact, detect other problems that can be caught earlier than they might be otherwise. And, if the living donor's remaining organ declines in function, that donor moves to the top of the transplant registry list.
In addition to considering living donation, we should all strongly consider donation after death. We should consider the question – does it make sense to bury or burn perfectly good corneas, heart valves, or tendons? This link has more information on how to become a donor.