Nerve regeneration has become on of the largest and most rigorously funded areas of medical research in history. This is largely due to the fact that every year, around 11,000 US citizens suffer spinal cord traumas or injuries that render them permanently disabled. The reason they're disabled is that unlike the peripheral nervous system, the central nervous system cannot regenerate damaged nerves. If it could, then spinal injuries could heal themselves.
The peripheral nervous system, also known as the PNS quite simply includes every nerve in the body outside of the brain, brain stem and spinal column. It is a far less complex system than the central nervous system which serves to bring information to the central nervous system about what's happening in the body and what's being sensed, as well as bring information from the central nervous system (CNS) to the body's muscles and carry out processes. Both the PNS and CNS are made up of nerves, which are much like electricity cables. If you cut one in half then the information can't pass through any more. This is what happens in some accidents, leaving individuals unable to walk, use their arms and do other more fundamental things like swallow or even breathe on their own.
If you damage a nerve in your hand, for example by cutting it badly with a knife, then it's quite likely that eventually it will heal, given the correct care and rapid treatment. This is because this nerve in the hand is part of the PNS and it is made up of 'Schwann' cells which are able to regenerate fully. The CNS, on the other hand, is made up of astrocyte and oligodendrocyte cells which do allow a certain amount of nerve growth, but at a crucial point terminate their efforts and as a result the process fails. The reason this happens is that these two types of cell actually inhibit nerve regeneration so that what does regenerate does not become abnormal. The CNS needs to be rigidly organized in order to work at all and, as seen in some skin repair that goes wrong (e.g. keloid scarring), the body's repair can go askew.
At present nerve regeneration for the central nervous system is unavailable, but not at all impossible. With the help of controversial stem cell research nerve regeneration is likely to become a possibility. Scientists are currently using growth factors, stem cells and guidance channels to research the possibilities and find a way of developing nerve regeneration for those who really need it. In a matter of decades or years a severe spinal injury needn't mean quadriplegia or paraplegia for the rest of a patient's lifetime.
At present, however, nerve regeneration remains a mystery for scientists. Yet it is not one that is far off being solved. With funding and support from health organizations the status of nerve regeneration research could be completely turned around.