There are three types of cells in the body, known as permanent, stable, and labile. Labile cells are those that divide constantly throughout our lives, such as the lining of our skin and gastrointestinal tracts. Stable cells are those that divide when needed, often cited with the regenerative capacities of the kidneys and liver. Permanent cells are those that do not divide, they are said to be trapped in the Go stage of the cell cycle. Brain and heart cells are said to be permanent. Despite the very famous animated TED talk video that claims your brain is completely new every 7 years, that is a completely false statement. Yes, your brain continues to have what is known as neuronal plasticity, creating and pruning synapses especially during childhood, and even to a certain extend throughout life, but they do not have the regenerative capacity that this video insinuates.
With the advent of cardiac transplants, and some success with therapeutic stem cell for the heart, there has been great progress. However, stem cells at present cannot be therapeutically used for brain injury (although this is illegally being done in India and China, the FDA has categorically banned it, there is still too much research to be done before this enters the clinical field.) I worked in a lab researching the therapeutic role of stem cells post traumatic brain injuries (TBI). After brain injury, there is some repair mediated through growth factors. We look for factors to both enhance those growth factors, and to inhibit the inhibitory factors. Here are links to the abstracts: