The objective of this chapter is to give the reader an understanding of monads and their associated algebraic structures, as well as some of their applications.

Monads first arose explicitly in homological algebra in the late 1950s and early 1960s. Comonads, i.e. monads in the dual of a category, first appear in Godement [15] who used the flabby sheaf comonad for resolving sheaves to compute sheaf co-homology. These structures were formalized and then applied to homotopy theory and homology theory by Huber, Barr, and Beck in [17, 4, 8]. In fact the history of the subject has been closely connected with applications. The applications to algebraic structures were initiated by the description of the algebras for a monad by Eilenberg and Moore [14] and closely related work of Kleisli [19].

Those mathematical structures which can be defined using only operations and equations have always attracted the mathematicians by their beauty, simplicity and wide range of applications. The first published proof that monads capture equational classes of algebras is the isomorphism theorem of Linton [23], extending Lawvere's description of algebraic theories [21]. Beck realized that monads describe universal algebra in the category of sets and proved his famous monadicity theorem [8].