The paper addresses the question whether in four spacetime dimensions,
besides standard supergravity theories, field theories exist whose symmetries
include local spacetime translations and supersymmetries generated by
transformations whose commutators contain infinitesimal local spacetime
translations. Several new supersymmetric free field theories are found which
may provide theories of that type. It is outlined how these free field theories
may be extended to globally supersymmetric theories with the desired
properties. One class of such theories is constructed explicitly. These
theories are similar to globally supersymmetric Yang-Mills theories in flat
spacetime but have supersymmetries generated by transformations whose
commutators generate infinitesimal general coordinate transformations with
field dependent gauge parameters.
