  A discussion of the number of degrees of freedom, and their dynamical
properties, in higher derivative gravitational theories is presented. The
complete non-linear sigma model for these degrees of freedom is exhibited using
the method of auxiliary fields. As a by-product we present a consistent
non-linear coupling of a spin-2 tensor to gravitation. It is shown that
non-vanishing $(C_{\mu\nu\alpha\beta})^2$ terms arise in $N=1$, $D=4$
superstring Lagrangians due to one-loop radiative corrections with light field
internal lines. We discuss the general form of quadratic $(1,1)$ supergravity
in two dimensions, and show that this theory is equivalent to two scalar
supermultiplets coupled to the usual Einstein supergravity. It is demonstrated
that the theory possesses stable vacua with vanishing cosmological constant
which spontaneously break supersymmetry. We then generalize this result to
$N=1$ supergravity in four dimensions. Specifically, we demonstrate that a
class of higher derivative supergravity theories is equivalent to two chiral
supermultiplets coupled in a specific way to Einstein supergravity. These
theories are shown to possess stable vacuum states with vanishing cosmological
constant which spontaneously break the $N=1$ supersymmetry.
