  We define a notion of category enriched over an oplax monoidal category $V$,
extending the usual definition of category enriched over a monoidal category.
Even though oplax monoidal structures involve infinitely many functors $V^n\to
V$, defining categories enriched over $V$ only requires the lower arity maps
$(n \leq 3)$, similarly to the monoidal case. The focal point of the enrichment
theory shifts, in the oplax case, from the notion of $V$-category (given by
collections of objects and hom-objects together with composition and unit maps)
to the one of categories enriched over $V$ (genuine categories equipped with
additional structures). One of the merits of the notion of categories enriched
over $V$ is that it becomes straightforward to define enriched functors and
natural transformations. We show moreover that the resulting 2-category
$\mathsf{Cat}_V$ can be put in correspondence (via the theory of distributors)
with the 2-category of modules over $V$. We give an example of such an enriched
category in the framework of operads: every cocomplete symmetric monoidal
category $C$ is enriched over the category of sequences in $C$ endowed with an
oplax monoidal structure stemming from the usual operadic composition product,
whose monoids are still the operads. As an application of the study of the
2-functor $V\mapsto\mathsf{Cat}_V$, we show that when $V$ is also endowed with
a compatible lax monoidal structure - thus forming a lax-oplax duoidal category
- the 2-category $\mathsf{Cat}_V$ inherits a lax 2-monoidal structure, thereby
generalising the corresponding result when the enrichment base is a braided
monoidal category. We illustrate this result by discussing the lax-oplax
structure on the category of $(R^\mathrm{e}, R^\mathrm{e})$-bimodules, whose
bimonoids are the bialgebroids. We also comment on the relations with other
enrichment theories (monoidal, multicategories, skew and lax).
