Accelerated Universe and gravity modified at largest observable distances

We discuss an idea that the cosmic acceleration of the Universe may be caused by modification of gravity at very large distances, and not by a dark energy source. Such a modification could, in particular, be triggered by extra space dimensions, to which gravity spreads over cosmic distances. This idea is testable by distant supernovae and other cosmological surveys. Also modified gravity predicts testable deviations in planetary motions. For the Moon this effect would show up in anomalous perihelion precessions of its orbit, and can be detected by lunar ranging experiments.