Let X be a smooth projective Berkovich space over a trivially or discretely valued field k of residue characteristic zero, and let L be an ample line bundle on X. We develop a theory of plurisubharmonic (or semipositive) metrics on L. In particular we show that the (non-Archimedean) Monge-Ampère operator induces a bijection between plurisubharmonic metrics and Radon probability measures of finite energy. In the discretely valued case, these results refine earlier work obtained in collaboration with C. Favre. In the trivially valued case, the results are new and will in subsequent work be shown to have ramifications for the study of K-stability.