Fundamental constants: from measurement to the universe, a window on gravitation and cosmology

Fundamental constants are a cornerstone of the physical laws. Any constant varying in space and/or time would signal a violation of local position invariance and be associated with a violation of the universality of free fall, and hence of the weak equivalence principle. It will reflect the existence of new degrees of freedom that couple to standard matter. Thus, testing for the stability of fundamental constants is of utmost importance for our understanding of gravity and for characterizing the domain of validity of General Relativity. Besides, it opens an independent window on dark matter and dark energy. As a consequence, thanks to the active development of experiments and of their accuracy, fundamental constants have become a key player in our search for physics beyond the standard model of and beyond General Relativity. This review details the roles of the fundamental constants in the laws of physics and in the construction of the International System of units. Then, the relations between constants, tests of the local position invariance and of the universality of free fall are exposed, as well as the construction of field theories that account for "varying constants". Then, the main experimental and observational constraints are described. It details the basics of each system, its dependence with respect to the primary parameters the variation of which can be constrained from observations, the known systematics effects and the most recent constraints. It also describes how these primary parameters can be related to the fundamental constants and the model-dependencies. Both time and space variation are considered. Given the huge increase of data and constraints, a general scheme to present experimental and observational results and to construct a collaborative data base that will be more efficient for the community and allow for better traceability, is proposed.