The observation of very large microwave-enhanced critical currents in superconductor-normal metal-superconductor (SNS) junctions at temperatures well below the critical temperature of the electrodes has remained without a satisfactory theoretical explanation for more than three decades. Here we present a theory of the supercurrent in diffusive SNS junctions under microwave irradiation based on the quasiclassical Green's function formalism. We show that the enhancement of the critical current is due to the energy redistribution of the quasiparticles in the normal wire induced by the electromagnetic field. The theory provides predictions across a wide range of temperatures, frequencies, and radiation powers, both for the critical current and the current-phase relationship.It was predicted by Eliashberg already in 1970 [1] that the condensation energy of a superconducting thin film can be increased by irradiating the film with microwaves. Within the framework of his theory, one can explain the microwave-induced increase of the critical current of superconducting bridges for temperatures very close to the critical temperature [2][3][4], which is known as the DayemWyatt effect. However, Eliashberg's mechanism fails to explain a related effect in diffusive SNS junctions. Several experiments have shown that upon irradiation the critical current can be enhanced by up to several orders of magnitude, even at temperatures well below the critical temperature of the superconducting electrodes [5,6]. Additionally, these experiments show that the critical current is a nonlinear function of the radiation power, which existing linear response theories [7,8] cannot explain.There is now renewed interest in this problem, triggered by recent experiments. Fuechsle et al. [9] measured the current-phase relationship under microwave irradiation, and reported that the current is progressively suppressed at phase differences close to π as the radiation amplitude increases. Moreover, Chiodi et al. [10] observed that critical current is enhanced when the microwave frequency is larger than the inverse diffusion time in the normal metal.To understand the microwave-assisted supercurrent in diffusive SNS junctions, we develop a microscopic theory based on the quasiclassical Keldysh-Usadel approach, which takes into account the nonlinear effects of the microwave irradiation. Our theory provides a quantitative description for a wide range of values of the temperature, microwave power, frequency, and the strength of inelastic scattering. In particular, we show that the large enhancement of the critical current originates from the presence of a minigap, E g , in the density of states of the normal wire. This minigap blocks some of the transitions caused by the microwave radiation, which results in a redistribution of quasiparticles, enhancing the supercurrent when the temperature T is comparable or larger than E g /k B . We also show that the nonequilibrium distribution in the normal wire leads to a highly non-sinusoidal currentphase relationship, ...