  We extend the theory of equivariant orthogonal spectra from finite groups to
profinite groups, and more generally from compact Lie groups to compact
Hausdorff groups. The G-homotopy theory is "pieced together" from the
G/U-homotopy theories for suitable quotient groups G/U of G; a motivation is
the way continuous group cohomology of a profinite group is built out of the
cohomology of its finite quotient groups. In the model category of equivariant
spectra Postnikov towers are studied from a general perspective. We introduce
pro-G-spectra and construct various model structures on them. A key property of
the model structures is that pro-spectra are weakly equivalent to their
Postnikov towers. We discuss two versions of a model structure with "underlying
weak equivalences". One of the versions only makes sense for pro-spectra. In
the end we use the theory to study homotopy fixed points of pro-G-spectra.
