Biological immortality
Biological immortality can be defined as the absence of a sustained increase in rate of mortality as a function of chronological age. A cell or organism that does not experience, or at some future point will cease aging, is biologically immortal.  However this definition of immortality was challenged in the new "Handbook of the Biology of Aging", because the increase in rate of mortality as a function of chronological age may be negligible at extremely old ages (late-life mortality plateau). But even though the rate of mortality ceases to increase in old age, those rates are very high (e.g., 50% chance of surviving another year at age 110 or 115 years of age).
There is no organism or individual cell that is literally immortal.  Any biologically "immortal" cell or organism can be killed by physical destruction.

# Cell lines
Biologists have chosen the word immortal to designate cells that are not limited by the Hayflick limit (where cells no longer divide because of DNA damage or shortened telomeres). (Prior to the work of Leonard Hayflick there was the erroneous belief fostered by Alexis Carrel that all normal somatic cells are immortal.)
The term immortalization was first applied to cancer cells that expressed the telomere-lengthening enzyme telomerase, and thereby avoided apoptosis (programmed cell death). Among the most commonly used cell lines are HeLa and Jurkat, both of which are immortalized cancer cells. Normal stem cells and germ cells can also be said to be immortal.
Immortal cell lines of cancer cells can be created by induction of oncogenes or loss of tumor suppressor genes.  One way to induce immortality is through viral-mediated induction of the large T-antigen, commonly introduced through simian virus 40 (SV-40).
In terms of multi-cellular organisms, immortality may not be a desirable condition, as the main controls over cancer are the apoptotic mechanisms.

# Bacteria
Bacteria can be said to be biologically immortal, but only as a colony. An individual bacterium can easily die. The two daughter bacteria resulting from cell division of a parent bacterium can be regarded as unique individuals or as members of a biologically "immortal" colony. The two daughter cells can be regarded as "rejuvenated" copies of the parent cell because damaged macromolecules have been split between the two cells and diluted. In the same way stem cells and gametes can be regarded as "immortal".

# Hydra
Hydras are a genus of simple, fresh-water animals possessing radial symmetry and no post-mitotic cells. The fact that all cells continually divide allows defects and toxins to be "diluted-away".  It has been suggested that hydras do not undergo senescence (aging), and so are biologically immortal. 

# Life extensionists
Some life extensionists, such as those who practice cryonics, have the hope that humans may someday become biologically immortal. This would not be the same as literal immortality, since people can always be murdered or die in accidents.  (Mind uploading, however, could allow literal immortality in a sense, by uploading backups into cloned or artificial bodies after an accident.  See Mind uploading in science fiction.)
Nanotechnology, and specifically of nanomedicine, have recently increased awareness of the possibilities for biological immortality in humans. A study published in Physiological and Biochemical Zoology in 2005 indicates that biological immortality may exist in humans at a late stage in life: "the exponential increase in age-specific death rate seemed to slow down considerably, if not cease."
The biogerontologist Aubrey de Grey of Cambridge University has proposed that damage to macromolecules, cells, tissues and organs could be repaired by advanced biotechnology. He calls his project SENS (Strategies for Engineered Negligible Senescence). Dr. de Grey has created the Methuselah Mouse Prize to award money to researchers who can significantly extend the lifespan of, or rejuvenate mice. It is his hope that if a monetary reward is offered, it will increase the likelihood of mice being rendered extremely, unnaturally long-lived by science, and that such a feat will inspire research into replicating this achievement in humans which may spark a "buy-in" with many people willing to pay large amounts of money for the benefits of such research.