Several general results are proved concerning the existence and uniqueness of various branched coverings of manifolds in dimensions 2 and 3. The results are applied to give a rather complete account as to which 3-manifolds are branched coverings of
S
3
{S^3}
,
S
2
×
S
1
{S^2}\, \times \,{S^1}
,
P
2
×
S
1
{P^2}\, \times \,{S^1}
, or the nontrivial
S
3
{S^3}
-bundle over
S
1
{S^1}
, and which degrees can be achieved in each case. In particular, it is shown that any closed nonorientable 3-manifold is a branched covering of
P
2
×
S
1
{P^2}\, \times \,{S^1}
of degree which can be chosen to be at most 6 and with branch set a simple closed curve. This result is applied to show that a closed nonorientable 3-manifold admits an open book decomposition which is induced from such a decomposition of
P
2
×
S
1
{P^2}\, \times \,{S^1}
.