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Gravitation as a fundamental interaction that governs all phenomena at largeand very small scales, but still not well understood at a quantum level, is acardinal missing link in unification of all physical interactions. Discovery ofthe present acceleration of the Universe, the dark matter and dark energyproblems are also a great challenge to modern physics, which may lead to a newrevolution in it. Integrable multidimensional models of gravitation andcosmology make up one of the proper approaches to studying basic issues andstrong field objects, the early and present Universe and black hole (BH)physics in particular. Our main results within this approach are described forboth cosmology and BH physics. Problems of the absolute G measurements and itspossible time and range variations are reflections of the unification problem.The choice, nature, classification and precision of determination of thefundamental physical constants as well as their role in a transition, expectedin 2011, to new definitions of the main SI units, supposed to be based onfundamental physical constants and stable quantum phenomena, are described. Theproblem of temporal variations of constants is also discussed, temporal andspatial variations of G in particular. A need for further absolute measurementsof G, its possible range and time variations is pointed out. The multipurposespace project SEE is briefly described, aimed at measuring G and its stabilityin space and time, with precision 3-4 orders better than at present. It mayanswer many important questions posed by gravitation, cosmology and unifiedtheories. A project of a laboratory experiment for testing possible deviationsfrom the Newton law is also presented.

Variations of constants as a test of gravity, cosmology and unified models