  We discuss collective excitations of a trapped dilute Fermi gas within a
hydrodynamic approximation. Analytical results are derived for both high- and
low-temperature limits and are applied to $^{40}$K and $^6$Li systems of
current experimental interest. We identify spectral signatures which can be
used to detect the onset of Fermi degeneracy. Also, we find an interesting
class of internal excitations with an unusual spectrum. Some of our results are
relevant to the case of trapped bosons as well. Our analysis suggests several
experiments which address fundamental problems of collective motion in quantum
fluids.
