Abstract: A generalized complex structure (in the sense of Hitchin) on a manifold is an endomorphism of $T\oplus T^*$ with square $-Id$, satisfying a certain integrability condition. Complex structures and symplectic structures yield natural generalized complex structures, and generalized complex geometry locally looks roughly like a product of complex and symplectic geometry. In the case of a hyperkahler manifold $M$, there is an $S^2\times S^2$-family of generalized complex structures compatible with the metric. We show that these structures can be assembled into a generalized complex structure on $Z=M\times S^2\times S^2$, which we call the ``generalized twistor space''. Developing a generalized twistor correspondence remains an open problem. This is joint work with Rebecca Glover.