We address the problem of nonequilibrium superconductivity in the presence of microwave irradiation. Using contemporary analytical methods, we refine the old Eliashberg theory and generalize it to arbitrary temperatures T and frequencies ω. Microwave radiation is shown to stimulate superconductivity in a bounded region in the (ω,T) plane. In particular, for T<0.47Tc and for ω>3.3kBTc superconductivity is always suppressed by a weak ac driving. We also study the supercurrent in the presence of microwave irradiation and establish the criterion for the critical current enhancement. Our results can be qualitatively interpreted in terms of the interplay between the kinetic ("stimulation" vs "heating") and spectral ("depairing") effects of the microwaves.