  A gauge invariant theory for electronic Raman scattering for superconductors
with anisotropic pairing symmetry is analyzed in detail. It is shown that Raman
scattering in anisotropic superconductors provides a wealth of
polarization-dependent information which probes the detailed angular dependence
of the superconducting ground state order parameter. The Raman spectra shows a
unique polarization dependence for various anisotropic pair- state symmetries
which affects the peak position of the spectra and generates symmetry dependent
low frequency and temperature power-laws which can be used to uniquely identify
the magnitude and symmetry of the energy gap. In particular, we calculate the
collective modes and the subsequent symmetry--dependent Raman spectra for a
$d_{x^{2}-y^{2}}$ superconductor and compare our results to the relevant data
on the cuprate systems as well as theoretical predictions for $s$--wave,
anisotropic $s$--wave and $s+id$ energy gaps. Favorable agreement is shown with
the predictions for $d_{x^{2}-y^{2}}$ pairing and the experimental data on
YBa$_{2}$Cu$_{3}$O$_{7}$, Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ and
Tl$_{2}$Ba$_{2}$CuO$_{6}$.
