In this thesis, we consider one of the most popular models of non-equilibrium statistical physics: the Asymmetric Simple Exclusion Process, in which particles jump stochastically on a one-dimensional lattice, between two reservoirs at fixed densities, with the constraint that each site can hold at most one particle at a given time. This model has the mathematical property of being integrable, which makes it a good candidate for exact calculations. What interests us in particular is the current of particles that flows through the system (which is a sign of it being out of equilibrium), and how it fluctuates with time. We present a method, based on the ``matrix Ansatz'' devised by Derrida, Evans, Hakim and Pasquier, that allows to access the exact cumulants of that current, for any finite size of the system and any value of its parameters. We also analyse the large size asymptotics of our result, and make a conjecture for the phase diagram of the system in the so-called ``s-ensemble''. Finally, we show how our method relates to the algebraic Bethe Ansatz, which was thought not to be applicable to this situation. \ \ noindent Keywords: Exclusion process, non-equilibrium statistical physics, large deviations, Bethe Ansatz.