  $N=2$ string theories are formulated in space-times with 2 space and 2 time
dimensions. If the world-sheet matter system consists of 2 chiral superfields,
the space-time is Kahler and the dynamics are those of anti-self-dual gravity.
If instead one chiral superfield and one twisted chiral superfield are used,
the space-time is a hermitian manifold with torsion and a dilaton. The string
spectrum consists of a scalar, which is a potential $K$ determining the metric,
torsion and dilaton. The dynamics imply that the curvature with torsion is
anti-self-dual, and an action is found for the potential $K$. It is argued that
any $N=2$ sigma-model with twisted chiral multiplets in any dimension can be
deformed to a conformally invariant theory if the lowest order contribution to
the conformal anomaly vanishes. If there are isometries, more general
geometries are possible in which the dilaton is the Killing potential for a
holomorphic Killing vector.
