Non-archimedean dynamics is the study of the limiting behaviour of points, measures and subsets under the iteration of an endomorphism of a non archimedean analytic space. The most studied case is the iteration of rational functions (and in particular polynomials) mapping the projective line onto itself. I will explain the basics of this type of study and expose my work on the question posed by Favre and Rivera-Letelier on whether periodic repelling points equidistribute to the canonical measure, which is well known to happen in the complex case. We will see that the answer is positive, at least in the case of polynomials, and that this type of problem is one of many instances where the Berkovich analytification proves fundamental in providing a theory analogous to the complex case.