Several versions of the pulsating (many-sheeted) model of the Universe are described, in particular, a model with reversal of the arrow of time. It is pointed out that the reversal point may be a singularity or may correspond to maximal cosmological expansion. The smoothing out of inhomogeneities and the growth of entropy produced by baryon decay are discussed, as well as processes involving black holes. It is conjectured that black holes are absent in the cosmological expansion-contraction cycle preceding the present one, and that such exceptional cycles do occur periodically.