  We extend Lurie's definition of enriched $\infty$-categories to notions of
left enriched, right enriched and bi-enriched $\infty$-categories, which
generalize the concepts of closed left tensored, right tensored and bitensored
$\infty$-categories and share many desirable features with them. We use
bi-enriched $\infty$-categories to endow the $\infty$-category of enriched
functors with enrichment that generalizes both the internal hom of the tensor
product of enriched $\infty$-categories when the latter exists, and the free
cocompletion under colimits and tensors. As an application we prove an end
formula for morphism objects of enriched $\infty$-categories of enriched
functors and compute the monad for enriched functors. We build our theory
closely related to Lurie's higher algebra: we construct an enriched
$\infty$-category of enriched presheaves via the enveloping tensored
$\infty$-category, construct transfer of enrichment via scalar extension of
bitensored $\infty$-categories, and construct enriched Kan-extensions via
operadic Kan extensions. In particular, we develop an independent theory of
enriched $\infty$-categories for Lurie's model of enriched $\infty$-categories.
