Explore a 32-minute lecture on twistor spaces and compact manifolds that can support both Kähler and non-Kähler complex structures. Delve into the historical context of this classical problem, tracing its origins to the 1950s with the work of Atiyah, Blanchard, and Calabi. Examine Tsanov's groundbreaking discovery in the 1980s of the first simply connected manifold exhibiting this dual nature - the twistor space of a K3 surface. Learn about the latest research demonstrating that the quaternion twistor space of a hyperkähler manifold shares this property, providing a higher-dimensional example of a simply connected manifold admitting both Kähler and non-Kähler complex structures.