  We introduce a generalization of monads, called relative monads, allowing for
underlying functors between different categories. Examples include
finite-dimensional vector spaces, untyped and typed lambda-calculus syntax and
indexed containers. We show that the Kleisli and Eilenberg-Moore constructions
carry over to relative monads and are related to relative adjunctions. Under
reasonable assumptions, relative monads are monoids in the functor category
concerned and extend to monads, giving rise to a coreflection between relative
monads and monads. Arrows are also an instance of relative monads.
