  In this paper we give a new foundational, categorical formulation for
operations and relations and objects parameterizing them. This generalizes and
unifies the theory of operads and all their cousins including but not limited
to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads,
their colored versions, as well as algebras over operads and an abundance of
other related structures, such as crossed simplicial groups, the augmented
simplicial category or FI--modules.
  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. This produces many new examples.
