  We consider the problem of prescribing the Gaussian and the geodesic
curvatures of a compact surface with boundary by a conformal deformation of the
metric. We derive some existence results using a variational approach, either
by minimization of the Euler-Lagrange energy or via min-max methods. One of the
main tools in our approach is a blow-up analysis of solutions, which in the
present setting can have diverging volume. To our knowledge, this is the first
time in which such an aspect is treated. Key ingredients in our arguments are:
a blow-up analysis around a sequence of points different from local maxima; the
use of holomorphic domain-variations; and Morse-index estimates.
