  In this text we combine the notions of supergeometry and supersymmetry. We
construct a special class of supermanifolds whose reduced manifolds are
(pseudo) Riemannian manifolds. These supermanifolds allow us to treat vector
fields on the one hand and spinor fields on the other hand as equivalent
geometric objects. This is the starting point of our definition of
{supersymmetric Killing structures}. The latter combines subspaces of vector
fields and spinor fields, provided they fulfill certain field equations. This
naturally leads to a superalgebra that extends the supersymmetry algebra to the
case of non-flat reduced space. We examine in detail the additional terms that
enter into this structure and we give a lot of examples.
