  A Killing submersion is a Riemannian submersion from an orientable 3-manifold
to an orientable surface whose fibers are the integral curves of a unit Killing
vector field in the 3-manifold. We classify all Killing submersions over
simply-connected Riemannian surfaces and give explicit models for many Killing
submersions including those over simply-connected constant Gaussian curvature
surfaces. We also fully describe the isometries of the total space preserving
the vertical direction. As a consequence, we prove that the only
simply-connected homogeneous 3-manifolds which admit a structure of Killing
submersion are the E(\kappa,\tau)-spaces, whose isometry group has dimension at
least 4.
