The thermodynamic formalism of uniformly hyperbolic dynamical systems is by now a fairly well understood subject. This is mainly because for such systems we have at our disposal a finite Markov partition. If we allow our ambient space to be non-compact, then (in general) we do not have a Markov partition, and some new methods are necessary to study its thermodynamic formalism. In this talk I will explain what is known about the thermodynamic formalism of the geodesic flow in a non-compact negatively curved manifold. In particular I will discuss the role of the entropy “at infinity” of the geodesic flow. I will put emphasis in the comparison with the situation for countable Markov shifts.