Monoids are one of the simplest theories in which we can compose elements of a set. Similarly, monads have been used extensively to treat composition of effectful code and its denotational semantics. During the last forty years the theory of monoids has been extended with diverse merge-like operators. In this article, we replicate several of these extensions at the level of monads. Building on a well-known relation between monads and monoids, we introduce monads with additional structure that account for merging. We show how monads with merging generalise and relate to models for well-known algebraic theories for concurrency such as classic process algebras and the more recent concurrent monoids. With these results, we aim to facilitate the generalisation and comparison of different approaches to concurrency.