  A Killing submersion is a Riemannian submersion from a 3-manifold to a
surface, both connected and orientable, whose fibres are the integral curves of
a Killing vector field, not necessarily unitary. The first part of this paper
deals with the classification of all Killing submersions in terms of two
geometric functions, namely the bundle curvature and the length of the Killing
vector field, which can be prescribed arbitrarily. In a second part, we show
that if the base is compact and the submersion admits a global section, then it
also admits a global minimal section. These turn out to be the only global
sections with constant mean curvature, which solves the Bernstein problem in
Killing submersions over compact base surfaces, as well as the Plateau problem
with empty boundary. Finally, we prove that any compact orientable stable
surface with constant mean curvature immersed in the total space of a Killing
submersion must be either an entire minimal section or everywhere tangent to
the Killing direction.
