This proceeding is based on a mini-course given at the Summer School "From kinetic equations to statistical mechanics" organized by the Centre Henri Lebesgue at Saint Jean de Monts from the 28th of June to the 2nd of July 2021. After recalling classical probabilistic limit theorems for sums of independent identically distributed random variables, we consider analogous results in a dynamical context. Motivated by examples coming from statistical mechanics, we are mostly interested in the Sinai billiard and in the Z 2-periodic Lorentz gas. We will also consider the Bunimovich stadium billiard and dispersive billiards with cusps. All these billiards are chaotic, with different behaviours. Additional explanations are given in the four independent appendices.