  The asymmetric simple exclusion process (ASEP) is a paradigm for
non-equilibrium physics that appears as a building block to model various
low-dimensional transport phenomena, ranging from intracellular traffic to
quantum dots. We review some recent results obtained for the system on a
periodic ring by using the Bethe Ansatz. We show that this method allows to
derive analytically many properties of the dynamics of the model such as the
spectral gap and the generating function of the current. We also discuss the
solution of a generalized exclusion process with $N$-species of particles and
explain how a geometric construction inspired from queuing theory sheds light
on the Matrix Product Representation technique that has been very fruitful to
derive exact results for the ASEP.
