  Supersymmetric field theories can be studied exactly on suitable off-shell
supergravity backgrounds. We show that in two dimensions such backgrounds are
identifiable with BRST invariant backgrounds of topological gravity coupled to
an abelian topological gauge multiplet. This latter background is required for
the consistent coupling of the topological `matter' YM theory to topological
gravity. We make use of this topological point of view to obtain, in a simple
and straightforward way, a complete classification of localizing supersymmetric
backgrounds in two dimensions. The BRST invariant topological backgrounds are
parametrized by both Killing vectors and $S^1$-equivariant cohomology of the
2-dimensional world-sheet. We reconstruct completely the supergravity
backgrounds from the topological data: some of the supergravity fields are
twisted versions of the topological backgrounds, but others are "composite",
i.e. they are non-linear functionals of them. We recover all the known
localizing 2-dimensional backgrounds and (infinitely) many more that have not
been explored so far. We show that the supersymmetric $\Omega$-deformation is
nothing but the background value of the ghost-for-ghost of topological gravity,
a result which holds for other dimensions too. The new localizing backgrounds
are characterized by non-trivial fluxes for both the graviphotons of the
supergravity multiplet.
