I study several kinds of generalizations of Montague’s theorem on the method of definition of a function by recursion on a well-founded relation. In particular, I extend Montague’s theorem to cover the following situations: recursion by a valuation system, non-well-founded recursion, and recursion on a dependence operator. By means of these extensions, several constructions employed in formal theories of truth and paradox can be recast as special applications of the generalized version of Montague’s theorem.