  We rework and generalize equivariant infinite loop space theory, which shows
how to construct $G$-spectra from $G$-spaces with suitable structure. There is
a classical version which gives classical $\Omega$-$G$-spectra for any
topological group $G$, but our focus is on the construction of genuine
$\Omega$-$G$-spectra when $G$ is finite. We also show what is and is not true
when $G$ is a compact Lie group.
  We give new information about the Segal and operadic equivariant infinite
loop space machines, supplying many details that are missing from the
literature, and we prove by direct comparison that the two machines give
equivalent output when fed equivalent input. The proof of the corresponding
nonequivariant uniqueness theorem, due to May and Thomason, works for classical
$G$-spectra for general $G$ but fails for genuine $G$-spectra. Even in the
nonequivariant case, our comparison theorem is considerably more precise,
giving an illuminating direct point-set level comparison.
  We have taken the opportunity to update this general area, equivariant and
nonequivariant, giving many new proofs, filling in some gaps, and giving a
number of corrections to results and proofs in the literature.
