Note: this is an unpublished manuscript.

Effects are a powerful and convenient component of programming. They enable programmers to interact with the user, take advantage of efficient stateful memory, throw exceptions, and nondeterministically execute programs in parallel. However, they also complicate every aspect of reasoning about a program or language, and as a result it is crucially important to have a good understanding of what effects are and how they work. In this paper we present a new framework for formalizing the semantics of effects that is more general and thorough than previous techniques while clarifying many of the important concepts. By returning to the category theoretic roots of monads, our framework is rich enough to describe the semantics of effects for a large class of languages including common imperative and functional languages. It is also capable of capturing more expressive, precise, and practical effect systems than previous approaches. Finally, our framework enables one to reason about effects in a language-independent manner, and so can be applied to many stages of language design and implementation in order to create more broadly applicable tools for programming languages.