Ever since the 1970's, holomorphic twistor spaces have been used to study the geometry and analysis of their base manifolds. In this talk, we will introduce integrable complex structures on twistor spaces fibered over complex manifolds that are equipped with certain geometrical data. The importance of these spaces will be shown to lie, for instance, in their applications to bihermitian geometry, also known as generalized Kahler geometry. (This is part of the generalized geometry program initiated by Nigel Hitchin.) By analyzing their twistor spaces, we will develop a new approach to studying bihermitian manifolds. In fact, we will demonstrate that the twistor space of a bihermitian manifold is equipped with two complex structures and natural holomorphic sections as well. This will allow us to construct tools from the twistor space that will lead, in particular, to new insights into the real and holomorphic Poisson structures on the manifold.