The non-perturbative study of quantum field theories and superstrings has shown the existence of surprising dualities, which exchange electricity and magnetism, at short and long distances behavior, weak and strong coupling constants, but which also provoke and appeal to mysterious objects ($M$-theory). The consequences in mathematics are manifold : mirror symmetry, special properties of automorphic forms, Seiberg-Witten invariants,... This exposition is meant to be an introduction, for mathematicians, to the idea of duality in physics.