Einstein's theory of General Relativity provides an elegant theory of gravity that has passed a large number of experimental tests. The main idea is that what we experience as gravity is a consequence of the curvature of spacetime itself. This curvature is produced by the presence of matter and Einstein's equations tell precisely how this happens. While for slow-moving objects and weak matter densities, Eintein's GR reduces to Newton's theory of gravity, GR predicts a variety of new phenomena. GR has passed a number of experimental tests, famously the precession of the perihelion of Mercury, the bending of light and the gravitational red-shift. It also predicts the existence of gravitational waves - wave-like solutions that are ripples in spacetime - and black holes. Gravitational waves were predicted by Einstein back in 1916, but remained a theoretical prediction until very recently. While they are produced by a variety of processes, their amplitudes are extremly small, thus making their direct detection difficult. Einstein's GR also allows us to study the universe on the largest scales - cosmology. This project starts with understanding the basic geometrical foundations of General Relativity, and can develop in a variety of ways: from a more mathematical approach to General Relativity with Riemannian Geometry, to a more practical oriented study with applications to cosmology, black holes, or gravitational waves and their generation and detection, or even alternate theories of gravity.