Cosmological models of the universe with reversal of time's arrow are considered. Formulations are given of the hypothesis of Cosmological CPT symmetry suggested earlier by the writer, and of the hypothesis of an open model with many sheets, with negative spatial curvature, and with possible violation of CTT symmetry by an invariant combined charge. The statistical paradox of reversibility is discussed for these models. The small dimensionless parameter $\delta_2/a_2$, which characterizes the mean spatial curvature of the universe, is explained as the result of the evolution of the universe through many successive cycles of expansion and contraction.