The celebrated Poincaré-Birkhoff theorem on area-preserving maps of the annulus is of fundamental importance in the fields of Hamiltonian dynamics and symplectic topology. In this talk I will formulate a twist condition, inspired by the Poincaré-Birkhoff theorem, which applies to the asymptotically linear Hamiltonian diffeomorphisms of Amann, Conley and Zehnder. When this twist condition is satisfied, together with some technical assumptions, the existence of infinitely many periodic points can be shown. In order to explain the key elements of the proof of this result, I will explore some of the features of Floer homology for asymptotically linear Hamiltonian diffeomorphisms.