Abstract. One the fundamental results in complex dynamics is the existence and uniqueness of the measure of maximal entropy. This measure has several interesting properties: it has exponential decay of correlations and satisfies the central limit theorem for Holder continuous observables, and furthermore it describes the asymptotic distribution of periodic points and of iterated preimages. In the arithmetic case it also describes the asymptotic distribution of points of small height. In the case of rational maps defined over an ultrametric field there is a measure with similar equidistribution properties. However some relatively simple examples show that this measure is not of maximal entropy, and that the topological entropy is not equal to the logarithm of the degree of the rational map. There are however some estimates that allow us to characterize those rational maps with zero topological entropy. The purpose of this talk is to describe the construction of this measure, and to show some of its fundamental properties. We will also see some key examples and open problems.