Let (C, l) be a stable curve with an involution. Following a classical construction one can define its Prym variety P, which in this case turns out to be a semiabelian group variety and usually not complete. In this paper we study the question whether there are "good" compactifications of P in analogy to compactified Jacobians. The answer to this question depends on whether we consider degenerations of principally polarized Prym varieties or degenerations with the induced (non-principal) polarization. We describe degeneration data of such degenerations. The main application of our theory lies in the case of degenerations of principally polarized Prym varieties where we ask whether such a degeneration depends on a given one-parameter family containing (C, l) or not. This allows us to determine the indeterminacy locus of the Prym map.