  In this paper we introduce a notion of {\it generalized operad} containing as
special cases various kinds of operad--like objects: ordinary, cyclic, modular,
properads etc. We then construct inner cohomomorphism objects in their
categories (and categories of algebras over them). We argue that they provide
an approach to symmetry and moduli objects in non-commutative geometries based
upon these "ring--like" structures. We give a unified axiomatic treatment of
generalized operads as functors on categories of abstract labeled graphs.
Finally, we extend inner cohomomorphism constructions to more general
categorical contexts. This version differs from the previous ones by several
local changes (including the title) and two extra references.
