We consider the classical problem of finding conformal metrics on a
surface such that both the Gaussian and the geodesic curvatures are
assigned functions.
We use variational methods and blow-up analysis to find existence of
solutions under suitable assumptions. A peculiar aspect of the problem
is that there are blow-up profiles with infinite volume that have to be 
excluded.
In order to do this, we classify their stability properties and employ
holomorphic vector fields. This is joint work with R. Lopez-Soriano and D. Ruiz.