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 naive version which gives naive G-spectra for any topological group G, but our focus is on the construction of genuine G-spectra when G is finite. 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 naive G-spectra for general G but fails hopelessly for genuine G-spectra when G is finite. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving a 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 some corrections to results in the literature.
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