| A Killing submersion is a Riemannian submersion from an orientable 3-manifold to an orientable surface, such that the fibres of the submersion are the integral curves of a Killing vector field without zeroes. The interest of this family of structures is that it yields a common treatment for a vast family of 3-manifolds, including, among others, the simply-connected homogeneous ones and the warped products with 1-dimensional fibres. In the first part of this talk we will give existence and uniqueness results for Killing submersions in terms of some geometric functions defined on the base surface. In the second part, we will consider constant mean curvature surfaces immersed in the total space of a Killing submersion which are transversal to the Killing vector field. This will take us to the classification of compact orientable stable surfaces with constant mean curvature immersed in such 3-manifolds. Dropping the stability condition, we will show that if the total space of a Killing submersion admits an immersed minimal sphere then the base surface is also a sphere. |