  We develop the theory of relative monads and relative adjunctions in a
virtual equipment, extending the theory of monads and adjunctions in a
2-category. The theory of relative comonads and relative coadjunctions follows
by duality. While some aspects of the theory behave analogously to the
non-relative setting, others require new insights. In particular, the universal
properties that define the algebra object and the opalgebra object for a monad
in a virtual equipment are stronger than the classical notions of algebra
object and opalgebra object for a monad in a 2-category. Inter alia, we prove a
number of representation theorems for relative monads, establishing the unity
of several concepts in the literature, including the devices of Walters, the
$j$-monads of Diers, and the relative monads of Altenkirch, Chapman, and
Uustalu. A motivating setting is the virtual equipment
$\mathbb{V}\text{-}\mathbf{Cat}$ of categories enriched in a monoidal category
$\mathbb{V}$, though many of our results are new even for $\mathbb{V} =
\mathbf{Set}$.
