We develop the parametrized post-Newtonian (PPN) formalism, which encompasses the weak-field, slow-motion regime, known as the post-Newtonian limit, of a wide range of metric theories of gravity. Ten PPN parameters are introduced, whose values depend upon the theory of gravity under study. We show that general properties of metric theories of gravity may be reflected in specific values of the PPN parameters, including the presence or absence of a preferred universal frame of reference, and the presence or absence of global conservation laws for energy, momentum and angular momentum.

We discuss the foundations of general relativity and all modern gravitational theories, based on the Einstein Equivalence Principle (EEP). We show that this principle is the basis for all metric theories of gravity including general relativity, in which gravity is a consequence of spacetime geometry. We review experimental test of the three pieces of EEP, the Weak Equivalence Principle, Local Lorentz Invariance and Local Position Invariance, and describe a number of general theoretical frameworks used to analyse EEP and its consequences.

We describe experimental tests of the effects of spinning bodies, which include precessions of spins as well as orbital perturbations. We give a technical and historical review of Gravity Probe B, a space experiment to measure the precession of orbiting gyroscopes, and the LAGEOS measurements of orbital perturbations, induced by the spinning Earth. We review experimental tests of post-Newtonian conservation laws, and the bounds on the relevant PPN parameters.

We describe tests of gravitational theory in the strong-field and dynamical regimes. Beginning with binary pulsars, we carry out an arrival-time analysis that reveals the relativistic effects on the time of arrival of radio pulses measured by an observer. We then describe the Hulse-Taylor binary pulsar, the double pulsar, a number of pulsars with white-dwarf companions, and the pulsar in a triple system, and describe the tests of gravitational theories that have been carried out using them. We describe the inspiral of compact binaries in general relativity and scalar-tensor theories, and the tests of gravitational theory that have been carried out using data from the gravitational wave detections of 2015 - 2017. We discuss future tests of general relativity in the strong-field regime, including tests using observations of stars orbiting the black hole at the center of the galaxy, tests involving accretion of matter onto black holes and neutron stars, and cosmological tests.

We derive the equations of motion for a variety of physical systems in the PPN formalism, including photons, fluid systems, and N-body systems consisting of well-separated self-gravitating objects. We also specialize to two-body systems and describe the framework for calculating perturbations of Keplerian orbit elements induced by post-Newtonian corrections in the equations of motion. For a class of theories based on an invariant action, we obtain the Lagrangian that describes the dynamics of an N-body system. We derive the locally-measured, or effective gravitational constant, as measured by a Cavendish experiment, within the PPN formalism. For spinning bodies, we obtain the equations of motion and the equations of spin precession.

We discuss the structure of compact bodies – neutron stars and black holes – in metric theories of gravity. We give a recipe for calculating the structure of such bodies, assumed to be non-rotating for simplicity, in generic metric theories, and then describe the results of calculations in specific theories. Turning to the motion of compact bodies, we develop the modified Einstein-Infeld-Hoffmann (EIH) framework, a generalization of the post-Newtonian N-body Lagrangian to systems containing one or more compact bodies, introducing parameters that may depend upon both the theory of gravity and the internal structure of each body. We analyse a number of observable effects in binary systems containing compact bodies and obtain their dependences on these parameters. The modified EIH parameters are calculated in a number of theories of gravity, including general relativity, scalar-tensor theories and Einstein-Aether/Khronometric theories.

We describe three central tests of general relativity, sometimes call the "classical" tests: the deflection of light, the Shapiro time delay and the perihelion advance of Mercury. After deriving each effect in detail within the PPN formalism, we describe the various measurements that lead to tight bounds on the relevant PPN parameters.