The total 2pN net shifts per orbit and the orbital precessions are calculated as the sum of two contributions: the direct ones due to the 2pN acceleration and the mixed, or indirect, ones caused by the 1pN instantaneous shifts during the orbital revolution. A comparison with other approaches existing in the literature is made.

The total 1pN gravitoelectric mass quadrupole orbital precessions of the Keplerian orbital elements are calculated in their full generality for an arbitrary orientation of the primary’s spin axis and a general orbital configuration of the test particle. Both the direct effects, due to the 1pN gravitoelectric mass quadrupole acceleration, and the mixed effects, due to the simultaneous action of the 1pN gravitoelectric mass monopole and Newtonian quadrupole accelerations, are calculated.

The impact of a wide range of post-Keplerian perturbing accelerations, of whatever physical origin, on different types of observation-related quantities (Keplerian orbital elements, anomalistic, draconitic, and sidereal orbital periods, two-body range and range rate, radial velocity curve and radial velocity semiamplitude of spectroscopic binaries, astrometric angles RA and dec., times of arrival of binary pulsars, characteristic timescales of transiting exoplanets along with their sky-projected spin-orbit angle) is analytically calculated with standard perturbative techniques in a unified and consistent framework. Both instantaneous and averaged orbital shifts are worked out to the first and second order in the perturbing acceleration. Also, mixed effects, due to the simultaneous action of at least two perturbing accelerations, are treated.

The precessions of the Keplerian orbital elements induced by several modified models of gravity are calculated. The latter ones are Yukawa, power-law, logarithmic, dark matter density profiles (exponential and power-law), once per revolution accelerations, constant accelerations, and Lorentz-violating symmetry.

The orbital precessions of the Keplerian orbital elements induced by the 1pN gravitomagnetic spin octupole moment of a rigidly rotating oblate spheroid are calculated in their full generality for an arbitrary orientation of the primary’s spin axis and a general orbital configuration of the test particle.

The impact of the 1pN gravitoelectric mass monopole acceleration, both in the test particle and in the two-body system of finite, comparable masses cases, is calculated for different types of observation-related quantities (Keplerian orbital elements, anomalistic, draconitic, and sidereal orbital periods, two-body range and range rate, radial velocity curve and radial velocity semiamplitude of spectroscopic binaries, astrometric angles RA and dec., times of arrival of binary pulsars, characteristic timescales of transiting exoplanets). The results are applied to a test particle orbiting a primary, a Sun–Jupiter exoplanet system, and to a S star in Sgr A*.

An overview of General Relativity is provided to a basic level. Its different nature with respect to the Newtonian Universal Gravitation is outlined. A cursory resume of the post-Newtonian approximation and its importance in testing Einstein’s theory is offered. A brief overview on the modified models of gravity that appeared in the last decades is outlined. A plan of the book is provided.