The goal of this article is to show how the study of metric spaces and of quantale-enriched categories interact, through generalization and specialization. We replace the recipient interval \([0,\infty ]\) of a (potentially \(\infty \) -valued) metric on a set by a commutative quantale, that is, by a complete lattice \(\mathcal{V}\) that comes with a commutative monoid operation interacting with the lattice structure. As the quantale \(\mathcal{V}\) may be seen as a thin cocomplete and symmetric monoidal-closed category, with arrows given by order, sets equipped with a generalized \(\mathcal{V}\) -valued metric will then become small \(\mathcal{V}\) -enriched categories. In this way, we explore how the theories of (not necessarily symmetric or separated) metric spaces and of enriched categories enhance each other. Our focus is on the enriched completion theory of these structures and on the important role that distributors between them play for them, and we offer a few glances at the many applications beyond the original metric theory, owing to the wide range of interesting quantales.