We propose a general theory of partial n-place operations based solely on the primitive notion of the application of a (possibly partial) operation to n objects. This theory is strongly selfdescriptive in that the fundamental manipulations of operations, that is, application, composition, abstraction, union, intersection and so on, are themselves internal operations. We give several applications of this theory, including implementations of partial n-ary λ-calculus, and other operation description languages. We investigate the issue of extensionality and give weakly extensional models of the theory.