  Gravitation as a fundamental interaction that governs all phenomena at large
and very small scales, but still not well understood at a quantum level, is a
cardinal missing link in unification of all physical interactions. Discovery of
the present acceleration of the Universe, the dark matter and dark energy
problems are also a great challenge to modern physics, which may lead to a new
revolution in it. Integrable multidimensional models of gravitation and
cosmology make up one of the proper approaches to studying basic issues and
strong field objects, the early and present Universe and black hole (BH)
physics in particular. Our main results within this approach are described for
both cosmology and BH physics. Problems of the absolute G measurements and its
possible time and range variations are reflections of the unification problem.
The choice, nature, classification and precision of determination of the
fundamental physical constants as well as their role in a transition, expected
in 2011, to new definitions of the main SI units, supposed to be based on
fundamental physical constants and stable quantum phenomena, are described. The
problem of temporal variations of constants is also discussed, temporal and
spatial variations of G in particular. A need for further absolute measurements
of G, its possible range and time variations is pointed out. The multipurpose
space project SEE is briefly described, aimed at measuring G and its stability
in space and time, with precision 3-4 orders better than at present. It may
answer many important questions posed by gravitation, cosmology and unified
theories. A project of a laboratory experiment for testing possible deviations
from the Newton law is also presented.
