  Branched covering have a long history from ramification of Riemann surfaces
to realization of 3-manifolds as covering ramified over a knots; from
geometrical topology to algebraic geometry. The present work investigates a
notion of branched covering "\`a la Fox" which is particularly natural for
(G,X)-manifolds. The work is two fold. First, we recall and enrich the current
state of the art (based upon Montesinos) and develop a Galois theory for such
branched covering together with a description of the fiber above branching
points. As a consequence, we solve two open questions of Montesinos and
construct an example related to another open question. Second, we present a
theory of singular (G,X)-manifolds and apply the theory of branched covering we
developped to extend the usual framework of (G,X)-manifolds to singular
(G,X)-manifolds; in particular, we construct a developping map for such
singular manifolds. An application to singular locally Minkowski manifolds is
given.
