The complete semilattices with complete semilattice homomorphisms form a symmetric monoidal closed category

$\mathcal{S}up$. A quantoloid

$\mathcal{Q}$ is a

$\mathcal{S}up$-enriched category. For a quantoloid

$\mathcal{Q}$, definitions of

$\mathcal{Q}$-enriched categories, of distributors and functors between

$\mathcal{Q}$-enriched categories are given. The theory of enriched categories is developed for

$\mathcal{Q}$-enriched categories. The notions and properties of adjoint functors, of Kan extensions, of weighted colimits and/or limits, of presheaves, of free cocompletion, of Cauchy completion, and of Morita equivalence are studied in

$\mathcal{Q}$-enriched categories for a quantoloid

$\mathcal{Q}$. Several examples illustrating obtained results are presented. The appendix is devoted to distributor calculus.