The asymmetric simple exclusion process is a model of interacting particles that appears in many realistic descriptions of low-dimensional transport with constraints and plays the role of a paradigm to understand the behaviour of non-equilibrium systems. The aim of this talk is to review some representative exact results about this model, to describe the methods involved and present some recent developments. In particular, by using the mathematical arsenal of integrable probabilities developed to solve the one-dimensional Kardar-Parisi-Zhang equation, we shall derive the exact finite-time distribution of a tagged particle in the symmetric simple exclusion process.