  Starting from the new minimal multiplet of supergravity in $2+2$ dimensions,
we construct two types of self-dual supergravity theories. One of them involves
a self-duality condition on the Riemann curvature and implies the equations of
motion following from the Hilbert-Einstein type supergravity action. The other
one involves a self-duality condition on a {\it torsionful} Riemann curvature
with the torsion given by the field-strength of an antisymmetric tensor field,
and implies the equations of motion that follow from an $R^2$-type action.
