  We present some new ideas about the possibility of life developing around
sub-giant and red giant stars. Our study concerns the temporal evolution of the
habitable zone. The distance between the star and the habitable zone, as well
as its width, increases with time as a consequence of stellar evolution. The
habitable zone moves outward after the star leaves the main sequence, sweeping
a wider range of distances from the star until the star reaches the tip of the
asymptotic giant branch. If life could form and evolve over time intervals from
$5 \times 10^8$ to $10^9$ years, then there could be habitable planets with
life around red giant stars. For a 1 M$_{\odot}$ star at the first stages of
its post main-sequence evolution, the temporal transit of the habitable zone is
estimated to be of several 10$^9$ years at 2 AU and around 10$^8$ years at 9
AU. Under these circumstances life could develop at distances in the range 2-9
AU in the environment of sub-giant or giant stars and in the far distant future
in the environment of our own Solar System. After a star completes its first
ascent along the Red Giant Branch and the He flash takes place, there is an
additional stable period of quiescent He core burning during which there is
another opportunity for life to develop. For a 1 M$_{\odot}$ star there is an
additional $10^9$ years with a stable habitable zone in the region from 7 to 22
AU. Space astronomy missions, such as proposed for the Terrestrial Planet
Finder (TPF) and Darwin should also consider the environments of sub-giants and
red giant stars as potentially interesting sites for understanding the
development of life.
