In this talk, I discuss some new classes of accelerating solutions to the standard cosmological problems (inflation, dark energy and late-time acceleration) from M theory compactification on compact internal manifold as homogeneous (and inhomogeneous) factor (and twister) spaces. This construction provides a certain realization of eternally accelerating universes from the time dependent compactification of higher dimensional gravities, other than that a scalar potential arising from slowly varying size of compact extra spaces may act as the dark energy of the universe. For a classical compactification of supergravities, the inflationary phase is normally observed only as a transient phenomenon. However, the possible non-perturbative effects and/or higher curvature effects could make this phase past-eternal, especially in the case of open inflation models.