Twistor forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We study twistor forms on compact Kähler manifolds and give a complete description up to special forms in the middle dimension. In particular, we show that they are closely related to Hamiltonian 2-forms. This provides the first examples of compact Kähler manifolds with non-parallel twistor forms in any even degree.