Precision gravity tests and the Einstein Equivalence Principle

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 its concepts, procedures and formalism. 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 describing, under the same standard, the metric and geodesic structure of the spacetime. 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 or not at quantum level. We are simply assuming its validity at fundamental scales. This question is crucial in any self-consistent theory of gravity. Furthermore, recent progress on relativistic theories of gravity, including deviations from General Relativity at various scales, such as extensions and alternatives to the Einstein theory, have to take into account, besides the Equivalence Principle, new issues like Dark Matter and Dark Energy, as well as the validity of fundamental principles like local Lorentz and position invariance. The general trend is that high precision experiments are conceived and realized to test both Einstein's theory and its alternatives at fundamental level using established and novel methods. For example, experiments based on quantum sensors (atomic clocks, accelerometers, gyroscopes, gravimeters, etc.) allow to set stringent constraints on well established symmetry laws (e.g. CPT and Lorentz invariance), 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. This provides us the ground for discussing the Equivalence Principle also in the framework of extended and alternative theories of gravity. In particular, we focus on the possibility to violate the Equivalence Principle at finite temperature, both in the frameworks of General Relativity and of modified gravity. Equivalence Principle violations in the Standard Model Extension are also discussed. 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.