In this paper we study solutions, possibly unbounded and sign-changing, of the Lane-Emden equationu = |u| p-1 u on unbounded domains of R N with N 2 and p > 1. We prove various classification theorems and Liouville-type results for C 2 solutions belonging to one of the following classes: stable solutions, finite Morse index solutions, solutions which are stable outside a compact set, radial solutions and non-negative solutions. Our results apply to subcritical, critical and supercritical values of the exponent p, and our analysis reveals the existence of a new critical exponent. This new critical exponent is larger than the classical critical exponent and, it depends on both the dimension N and the geometry of the considered unbounded domain. Some results about the qualitative properties of solutions, in arbitrary domains of R N , are also obtained. In particular, we prove a universal a priori estimate for stable solutions in arbitrary proper domains and study the behaviour of a stable solution near an isolated singularity. Applications to bounded domains are also considered. Many of our results are sharp.