We introduce a generalisation of monads, called relative monads, allowing for underlying functors between difeerent 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 core ection between relative monads and monads. Arrows are also an instance of relative monads.