  General Relativity is today the best theory of gravity addressing a wide
range of phenomena. Our understanding of physical laws, from cosmology to local
scales, cannot be properly formulated without taking into account it. It is
based on one of the most fundamental principles of Nature, the Equivalence
Principle, which represents the core of the Einstein theory of gravity. The
confirmation of its validity at different scales and in different contexts
represents one of the main challenges of modern physics both from the
theoretical and the experimental points of view. A major issue related to this
principle is the fact that we actually do not know if it is valid at quantum
level. Furthermore, recent progress on relativistic theories of gravity have to
take into account new issues like Dark Matter and Dark Energy, as well as the
validity of fundamental principles like local Lorentz and position invariance.
Experiments allow to set stringent constraints on well established symmetry
laws, on the physics beyond the Standard Model of particles and interactions,
and on General Relativity and its possible extensions. In this review, we
discuss precision tests of gravity in General Relativity and alternative
theories and their relation with the Equivalence Principle. In the first part,
we discuss the Einstein Equivalence Principle according to its weak and strong
formulation. We recall some basic topics of General Relativity and the
necessity of its extension. Some models of modified gravity are presented in
some details. The second part of the paper is devoted to the experimental tests
of the Equivalence Principle in its weak formulation. We present the results
and methods used in high-precision experiments, and discuss the potential and
prospects for future experimental tests.
