  In this paper, we survey the nature of spinors and supersymmetry (SUSY) in
various types of spaces. We treat two distinct types of spaces: flat spaces and
spaces of constant (non-zero) curvature. The flat spaces we consider are either
three or four dimensional of signatures 3 + 1, 4 + 0, 2 + 2 and 3 + 0. In each
of these cases, SUSY generators anti-commute to yield the generators of
translations in the non-compact flat spaces. The spaces of constant curvature
we consider are two-dimensional: the surface of the sphere S_2 and the
Anti-deSitter space AdS_2. S_2 is embedded in a 3 + 0 Euclidean space while
AdS_2 is embedded in 2 + 1 Minkowski space. The SUSY generators in these cases
anti-commute to yield the generators of the isometry groups (SO(3) or SO(2,1))
of the space involved. We also report on some recent developments in looking
for superspace realizations of these SUSY algebras. We can report good progress
in the 3 + 0 Euclidean and in the AdS_2 case, somewhat less in the S_2 case. In
each of the compact cases, we can construct field multiplet models carrying
invariance under the full SUSY algebra.
