We consider critical and supercritical Liouville equations on surfaces and on domains of \mathbb{R}^2 under Dirichlet boundary conditions. Using some tools of the "critical point theory at Infinity" of A. Bahri, we derive new existence and multiplicity results. In particular we provide an Euler-Hopf type criterium for the existence of solutions in the so called "resonant case" when the involved parameter is a multiple of 8 \pi. Such a criterium can be seen as a generalization of celebrated the degree formula of C.C.Chen and C.S. Lin.