Linear Weingarten surfaces in Euclidean and hyperbolic space

In this paper we review some author’s results about Weingarten surfaces in Euclidean space \r^3 and hyperbolic space ${\text{h}}^3$. 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 ${\text{h}}^3$ that are invariant by a group of parabolic isometries, obtaining its classification.