  Let X be a smooth complex surface of general type such that the image of the
canonical map $\phi$ of X is a surface $\Sigma$ and that $\phi$ has degree
$\delta\geq 2$. Let $\epsilon\colon S\to \Sigma$ be a desingularization of
$\Sigma$ and assume that the geometric genus of S is not zero. Beauville has
proved that in this case S is of general type and $\epsilon$ is the canonical
map of S. Beauville has also constructed the only infinite series of examples
$\phi:X\to \Sigma$ with the above properties that was known up to now. Starting
from his construction, we define a {\em good generating pair}, namely a pair
$(h:V\to W, L)$ where h is a finite morphism of surfaces and L is a nef and big
line bundle of W satisfying certain assumptions. We show that by applying a
construction analogous to Beauville's to a good generating pair one obtains an
infinite series of surfaces of general type whose canonical map is 2-to-1 onto
a canonically embedded surface. In this way we are able to construct more
infinite series of such surfaces. In addition, we show that good generating
pairs have bounded invariants and that there exist essentially only 2 examples
with $\dim |L|>1$. The key fact that we exploit for obtaining these results is
that the Albanese variety P of V is a Prym variety and that the fibre of the
Prym map over P has positive dimension.
