  We set up operadic foundations for equivariant iterated loop space theory. We
start by building up from a discussion of the approximation theorem and
recognition principle for V-fold loop G-spaces to several avatars of a
recognition principle for infinite loop G-spaces. We then explain what genuine
permutative G-categories are and, more generally, what E_{\infty} G-categories
are, giving examples showing how they arise. As an application, we prove the
equivariant Barratt-Priddy-Quillen theorem as a statement about genuine
G-spectra and use it to give a new, categorical, proof of the tom Dieck
splitting theorem for suspension G-spectra. Other examples are geared towards
equivariant algebraic K-theory.
