  In this paper we review some author's results about Weingarten surfaces in
Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the
search of examples of linear Weingarten surfaces that satisfy a certain
geometric property. First, we consider Weingarten surfaces in $\r^3$ that are
foliated by circles, proving that the surface is rotational, a Riemann example
or a generalized cone. Next we classify rotational surfaces in $\r^3$ of
hyperbolic type showing that there exist surfaces that are complete. Finally,
we study linear Weingarten surfaces in $\h^3$ that are invariant by a group of
parabolic isometries, obtaining its classification.
