[ed note - this teshuva has been in the news recently due to the recent discussions concerning the RCA's paper about brain death. I am not an expert in the issues involved, nor am I seeking to contribute to the overall discussion. However, I do feel that in any overheated debate, it is important to actually check the sources that are the source of the issue.]
In a teshuva from 1970, Rav Moshe Feinstein was asked to rule on the medical notion of brain death. He begins by stating that life is determined by breathing, and cessation of brain activity does not necessarily mean cessation of respiratory activity. As such, brain death would not be considered death [ed note - Rav Moshe's son-in-law, Rabbi Dr. Moshe Tendler, has drawn a distinction with regard to this teshuva between brain death and cerebral death - see here]. So long as the doctors are able to detect some signs of life, even if they require sensitive high-tech equipment to do so, they should treat the patient to the utmost degree in order to save him.
[Rav Moshe pauses at this point to note that we generally are not concerned with things that can only be detected via microscopes and other instruments which perceive that which we cannot perceive with our senses. The case of a patient whose heartbeat can only be detected via electric radiogram is an exception to this rule.]
The next extensive part of this teshuva deals with the various parts of the body that are discussed in sections of the Gemara that discuss determining death, such as the nose and the navel. Rav Moshe concludes that neither one is actually responsible for life, but rather are merely locations on the body where we can perceive whether or not life functions are continuing, and ultimately he leans towards the view of the Chacham Tzvi that the heart is the major determinant of life.
At the very end of the teshuva, Rav Moshe discusses the issue of heart transplants, which had been done for the first time in 1967, and was still exceedingly rare at the time of this teshuva (see here for more history). Rav Moshe feels that the removal of the heart from the recipient is tantamount to murder, and that doctors are not trustworthy that the transplant will work both because who knows whether the new heart will work any better than the old one, and because past recipients have lived only a short time. It would be interesting to see more recent literature on this topic as medical knowledge in this field has improved vastly over the past four decades. Readers are encouraged to send in references.