  High energy electron-photon colliders provide unique opportunities for
probing physics beyond the standard model. We have studied the experimental
signatures for two supersymmetric scenarios, with the lightest supersymmetric
particle (LSP) being either the lightest neutralino or the gravitino. In the
``neutralino LSP'' scenario favored by the minimal supersymmetric standard
model (MSSM), it is found that some basic parameters of the model, $\mu$,
$\tan\beta$, $M_1$ and $M_2$, may be uniquely determined from the outgoing
electron energy spectrum without assuming high scale unification of the masses
or couplings. In the ``gravitino LSP'' scenario which occurs naturally in
models of low energy dynamical supersymmetry breaking, it is possible to have
background-free signatures if the next-to-lightest supersymmetric particle
(NLSP) has a long decay length. In cases that the NLSP decays quickly, ways to
distinguish among the experimental signatures of the two scenarios and of the
standard model (SM) background are discussed.
