The correlation between interpolation theorems of logic and certain properties of the class of models related to the amalgamation property is well known. In classical sentential and first-order logic it takes the form of a correspondence between Craig's interpolation theorem and Robinson's joint consistency lemma. In the algebraic versions of these logics the joint consistency property can be replaced by the amalgamation property for Boolean algebras and locally finite cylindric algebras. The connection between interpolation and amalgamation has also been explored in the context of intermediate and modal logics, equational logic, and general deductive systems in the sense of Tarski. More recently, logical interpolation results have been shown to have interesting applications for the specification of abstract data types, especially with regard to the important problem of modularization. In the present paper we take advantage of the new domain of abstract algebraic logic to present a unified theory of interpolation, joint consistency, model extension, and amalgamation that comprehends all these results. In this general context the Craig interpolation property ramifies into several different interpolation-like properties, one of which is closely related to the familiar congruence extension property of universal algebra. We show how under quite weak conditions on the logical system, each interpolation property is equivalent to an extension or amalgamation-type property of the appropriate model class.