In this paper we give a new foundational categorical formulation for operations and relations and objects parameterizing them. This generalizes operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads as well as algebras over operads and an abundance of other related structures, such as FI--algebras.  The usefulness of this approach is that it allows us to handle all the classical as well as more esoteric structures under a common framework and we can treat all the situations simultaneously. Many of the known constructions simply become Kan extensions.  In this common framework, we also derive universal operations, such as those underlying Deligne's conjecture, construct Hopf algebras as well as perform resolutions, (co)bar transforms and Feynman transforms which are related to master equations. For these applications, we construct the relevant model category structures.