We discuss possible variations of the effective gravitational constant with length scale, predicted by most of alternative theories of gravity and unified models of physical interactions. After giving a brief general exposition, we review in more detail the predicted corrections to Newton's law of gravity in diverse brane world models. We consider various configurations in 5 dimensions (flat, de Sitter and AdS branes in Einstein and Einstein-Gauss-Bonnet theories, with and without induced gravity and possible incomplete graviton localization), 5D multi-brane systems and some models in higher dimensions. A common feature of all models considered is the existence of corrections to Newton's law at small radii comparable with the bulk characteristic length: at such radii, gravity on the brane becomes effectively multidimensional. Many models contain superlight perturbation modes, which modify gravity at large scale and may be important for astrophysics and cosmology.

Electric S-brane solutions with two non-composite electric branes and a set of l scalar fields are considered. The intersection rules for branes correspond to Lie algebras A 2 , C 2 and G 2 . The solutions contain five factor spaces. One of them, M 0 , is interpreted as our 3dimensional space. It is shown that there exists a time interval where accelerated expansion of our 3-dimensional space is compatible with a small enough variation of the effective gravitational constant G(τ ). This interval contains τ 0 , a point of minimum of the function G(τ ). A special solution with two phantom scalar fields is analyzed and it is shown that in the vicinity of the point τ 0 the time variation of G(τ ) (calculated in the linear approximation) decreases in the sequence of Lie algebras A 2 , C 2 and G 2 .

We consider multidimensional gravity with a Lagrangian containing the Ricci tensor squared and the Kretschmann invariant. In a Kaluza-Klein approach with a single compact extra space of arbitrary dimension, with the aid of a slow-change approximation (as compared with the Planck scale), we build a class of spatially flat cosmological models in which both the observed scale factor a(τ ) and the extra-dimensional one, b(τ ), grow exponentially at large times, but b(τ ) grows slowly enough to admit variations of the effective gravitational constant G within observational limits. Such models predict a drastic change in the physical laws of our Universe in the remote future due to further growth of the extra dimensions.