The first goal of this paper was to survey my definition in [19] of measures on non-archimedean analytic spaces in the sense of Berkovich and to explain its applications in Arakelov geometry. These measures are analogous the measures on complex analytic spaces given by products of first Chern forms of hermitian line bundles. In both contexts, archimedean and non-archimedean, they are related with Arakelov geometry and the local height pairings of cycles. However, while the archimedean measures lie at the ground of the definition of the archimedean local heights in Arakelov geometry, the situation is reversed in the ultrametric case: we begin with the definition of local heights given by arithmetic intersection theory and define measures in such a way that the archimedean formulae make sense and are valid. The construction is outlined in Section 1, with references concerning metrized line bundles and the archimedean setting. More applications to Arakelov geometry and equidistribution theorems are discussed in Section 3.

The relevance of Berkovich spaces in Diophantine geometry has now made been clear by many papers; besides [19] and [20] and the general equidistribution theorem of Yuan [59], I would like to mention the works [38, 39, 40, 30] who discuss the function field case of the equidistribution theorem, as well as the potential theory on non-archimedean curves developed simultaneously by Favre, Jonsson & Rivera-Letelier [32, 33] and Baker & Rumely for the projective line [8], and in general by A. Thuillier's PhD thesis [55].