  We study homotopy-coherent commutative multiplicative structures on
equivariant spaces and spectra. We define N-infinity operads, equivariant
generalizations of E-infinity operads. Algebras in equivariant spectra over an
N-infinity operad model homotopically commutative equivariant ring spectra that
only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the
operad. Analogously, algebras in equivariant spaces over an N-infinity operad
provide explicit constructions of certain transfers. This characterization
yields a conceptual explanation of the structure of equivariant infinite loop
spaces.
  To explain the relationship between norms, transfers, and N-infinity operads,
we discuss the general features of these operads, linking their properties to
families of finite sets with group actions and analyzing their behavior under
norms and geometric fixed points. A surprising consequence of our study is that
in stark contract to the classical setting, equivariantly the little disks and
linear isometries operads for a general universe $U$ need not determine the
same algebras.
  Our work is motivated by the need to provide a framework to describe the
flavors of commutativity seen in recent work of the second author and Hopkins
on localization of equivariant commutative ring spectra.
