The study of large deviations has emerged as a major theme of research in statistical physics over the last two decades, with multiple applications in several inter-disciplinary fields such as nonequilibrium physics, climate science, information theory, disordered systems, etc. This Les Houches Summer School, organized by A. Dhar, J. Krug, S.N. Majumdar, A. Rosso and G. Schehr, has gathered a number of international experts, spanning across disciplines, who provided a broad overview of this rapidly evolving field.
In this set of two lectures (1.5h each) I discuss the statistics of current fluctuations in many-body nonequilibrium systems, using both macroscopic fluctuation theory and microscopic spectral methods. Particular emphasis is put on describing the optimal path leading to a rare event, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level.