  Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an
oriented G-equivariant vector bundle. One defines the spaces of equivariant
forms with generalized coefficients on V and M. An equivariant Thom form
$\theta$ on V is a compactly supported closed equivariant form such that its
integral along the fibres is the constant function 1 on M. Such a Thom form was
constructed by Mathai and Quillen. Its restriction to M gives a representative
of the equivariant Euler class of V.
  In the supergeometric situation we give proper definitions of all the objects
involved. But, in this case a Thom form doesn't always exist. In this article,
when the action of G on V is sufficiently non-trivial, we construct such a Thom
form with generalized coefficients. We use it to construct an equivariant Euler
form of V and to generalize Berline-Vergne's localization formula to the
supergeometric situation.
