There has been some exciting work regarding the growing or re-growing of human body parts.
Some animals have an incredible ability to regenerate missing body parts—a classic example being some species of starfish. However, for the most part, it is not possible to regrow complex organs. In humans, damaged tissue usually is replaced by “generic” scar tissue, if it all. There are several reasons why humans cannot regenerate most body parts. For one, once cells become specialized, they often lost the ability to divide. Another reason is that arms and hearts and so on develop according to a specific pattern during embryonic development in the womb; there is no “program” for starting with part of a fully-developed structure and regrowing the rest. And furthermore, there is some evidence that the ability to regenerate has been sacrificed to avoid cancer. Cancer is essentially uncontrolled cell growth; many checkpoints that help regulate this may also prevent stem cells from recreating damaged tissue. Presumably, this balance reflects an optimum balance for the survival of our ancestors.
But scientists are working to augment this ability. As New Scientist reports, researchers in Japan were able to grow tooth buds in the laboratory, then transplant them to the jaws of mice where they developed into normal teeth; they even developed a blood supply. Also reported in New Scientist is the efforts of American researchers to grow new ligaments in the laboratory. Ligament injuries are quite common, and they tend to heal quite poorly.
Finally, LiveScience discusses regrowth of human fingertips after accidents, and mentions a few notable cases.