  In this paper we consider the Prym variety $P(\widetilde{C}/C)$ associated to
a Galois coverings of curves $f:\widetilde{C}\to C$ branched at $r$ points. We
discuss some properties and equivalent definitions and then consider the Prym
map $\mathcal{P}=\mathcal{P}(G,g,r):R(G,g,r)\to A_{p,\delta}$ with $\delta$ the
type of the polarization. For Galois coverings whose Galois group is abelian
and metabelian (non-abelian) we show that the differential of this map at
certain points is injective. We also consider the Abel-Prym map
$u:\widetilde{C}\to P(\widetilde{C}/C)$ and prove some results for its
injectivity. In particular we show that in contrast to the classical and cyclic
case, the behavior of this map here is more complicated. The theories of
abelian and metabelian Galois coverings play a substantial role in our analysis
and have been used extensively throughout the paper.
