Abstract: We discuss a conjecture of Kotschick which predicts that a compact Kähler manifold admits a holomorphic one-form without zeros if and only if the underlying real manifold admits a smooth fibration over the circle. In particular, we will explain that this conjecture holds for smooth projective threefolds (joint work with Feng Hao) and that it is closely related to a Conjecture of Bobadilla?Kollár which predicts that a proper map between complex manifolds is submersive if and only if it is a homological fibre bundle (joint with Ruijie Yang).