Abstract In this chapter we explain how particle scattering can be described by local observables. First we introduce the concept of the S-matrix, which is an idea common to classical and quantum mechanics. We show the construction of incoming and outgoing states which have a certain asymptotic behaviour at time −∞ and +∞, and we also show that the constructed states have an acceptable physical asymptotic behaviour. Then we derive the representation of the S-matrix given by these scattering states and discuss its properties.