  This paper is devoted to the exploration of some of the geometrical issues
raised by the $N=2$ superstring. We begin by reviewing the reasons that
$\beta$-functions for the $N=2$ superstring require it to live in a
four-dimensional self-dual spacetime of signature $(--++)$, together with some
of the arguments as to why the only degree of freedom in the theory is that
described by the gravitational field. We then move on to describe at length the
geometry of flat space, and how a real version of twistor theory is relevant to
it. We then describe some of the more complicated spacetimes that satisfy the
$\beta$-function equations. Finally we speculate on the deeper significance of
some of these spacetimes.
