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 β functions for the N = 2 superstring require it to live in a four-dimensional self-dual space–time 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 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 space–times that satisfy the β function equations. Finally we speculate on the deeper significance of some of these space–times.