  We address the question of when a covering of the boundary of a surface can
be extended to a covering of the surface (equivalently: when is there a
branched cover with a prescribed monodromy). If such an extension is possible,
when can the total space be taken to be connected? When can the extension be
taken to be regular? We give necessary and sufficient conditions for both
finite and infinite covers (infinite covers are our main focus). In order to
prove our results, we show group-theoretic results of independent interests,
such as the following extension (and simplification) of the theorem of Ore}:
every element of the infinite symmetric group is the commutator of two elements
which, together, act transitively
