In 1687 Newton published his work on the universal law of gravity in his book Philosophiae Naturalis Principia Mathematica ( Latin:Mathematical Principles of Natural Philosophy). Newton's law of gravitation states that: every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. If the particles have masses m1 and m2 and are separated by a distance r (from their centers of gravity), the magnitude of this gravitational force.
F is the magnitude of the gravitational force between the two point masses
G is the gravitational constant
m1 is the mass of the first point mass
m2 is the mass of the second point mass
r is the distance between the two point masses

Newton's conception and quantification of gravitation held until the beginning of the 20th century, when the German-born physicist Albert Einstein proposed the general theory of relativity. In this theory Einstein proposed that inertial motion occurs when objects are in free-fall instead of when they are at rest with respect to a massive object such as the Earth (as is the case in classical mechanics). The problem is that in flat spacetimes such as those of classical mechanics and special relativity, there is no way that inertial observers can accelerate with respect to each other, as free-falling bodies can do as they each are accelerated towards the center of a massive object.

To deal with this difficulty, Einstein proposed that spacetime is curved by the presence of matter, and that free-falling objects are following the geodesics of the spacetime. More specifically, Einstein discovered the field equations of general relativity, which relate the presence of matter and the curvature of spacetime. The Einstein field equations are a set of 10 simultaneous, non-linear, differential equations whose solutions give the components of the metric tensor of spacetime. This metric tensor allows to calculate not only angles and distances between space-time intervals (segments) measured with the coordinates against which the spacetime manifold is being mapped but also the affine-connection from which the curvature is obtained, thereby describing the spacetime's geometrical structure. Notable solutions of the Einstein field equations include:

The Schwarzschild solution, which describes spacetime surrounding a spherically symmetric non-rotating uncharged massive object. For compact enough objects, this solution generated a black hole with a central singularity.
The Reissner-Nordström solution, in which the central object has an electrical charge. For charges with a geometrized length which are less than the geometrized length of the mass of the object, this solution produces black holes with two event horizons.
The Kerr solution solution for rotating massive objects. This solution also produces black holes with multiple event horizons.
The cosmological Robertson-Walker solution, which predicts the expansion of the universe.
General relativity has enjoyed much success because of how its predictions have been regularly confirmed. For example:

General relativity accounts for the anomalous precession of the planet Mercury.
The prediction that time runs slower at lower potentials has been confirmed by the Pound-Rebka experiment, the Hafele-Keating experiment, and the GPS.
The prediction of the deflection of light was first confirmed by Arthur Eddington in 1919, and has more recently been strongly confirmed through the use of a quasar which passes behind the Sun as seen from the Earth. See also gravitational lensing.
The time delay of light passing close to a massive object was first identified by Shapiro in 1964 in interplanetary spacecraft signals.
Gravitational radiation has been indirectly confirmed through studies of binary pulsars.
The expansion of the universe (predicted by the Robertson-Walker metric) was confirmed by Edwin Hubble in 1929.