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