  We address the problem of non-equilibrium superconductivity in the presence
of microwave irradiation. We refine the old Eliashberg theory and generalize it
to arbitrary temperatures $T$ and frequencies $\omega$. Microwave radiation is
shown to stimulate superconductivity in a bounded region in the $(\omega,T)$
plane. In particular, for $T<0.47\, T_c$ and for $\hbar\omega>3.3\, k_BT_c$
superconductivity is always suppressed by a weak \emph{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.
