Twistor spaces and compact manifolds admitting both Kähler and non-Kähler structures

In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950's. In the 1980's Tsanov gave an example of a simply connected manifold that admits both Kähler and non- Kähler complex structures - the twistor space of a K3 surface. Here we show that the quaternion twistor space of a hyperkähler manifold has the same property.