This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce operational monads, thus covering new applications such as the-calculus, Positive GSOS specifications, and the big-step, simply-typed, call-by-value-calculus. Finally, we design a notion of signature for operational monads that covers all our examples.