General relativity is our best description of how gravity works on a fundamental level: spacetime is intrinsically curved, and “gravity” is how we perceive matter evolving on this curved spacetime. The curvature, on the other hand, is determined precisely by the matter present in spacetime. As summarized by John A. Wheeler, spacetime tells matter how to move, while matter tells spacetime how to curve. Not only is general relativity an experimental success, but it is also an extremely rich mathematical theory.
One of the most famous predictions of general relativity is the existence of black holes. Black holes are regions of spacetime where the curvature is so intense nothing can escape them. Astrophysically, these regions can be formed in the gravitational collapse of very large stars, which become so compact that the gravitational field (and hence the curvature) becomes enormous. While these regions were originally regarded with disbelief, nowadays we have strong experimental evidence of their existence, such as the images obtained by the EHT collaboration.
A second important prediction of general relativity is the existence of gravitational waves. The motion of matter on spacetime can lead to the formation of “ripples” in spacetime, which propagate at the speed of light. These ripples are capable of distorting space and time where they pass, and this allows us to detect them on Earth. This is done, for example, by the LIGO collaboration.
General relativity can also be used to study the universe as a whole, which is the basis for modern cosmology. We now know much of the history of the universe and how it evolved. For example, it is expanding at an accelerated rate and it started in a very hot and dense state.
The mathematical aspects of the theory are also very rich. The use of ideas from differential topology and differential geometry (which are indeed the basic language for general relativity) thrived in several interesting results. For example, the singularity theorems, which established that the occurrence of “punctures” (singularities, in the jargon) in spacetime is a robust prediction of the strong field regime. For instance, singularities occur in the interior of black holes and at the beginning of the universe (the Big Bang). In these regions, the curvatures are too large for classical general relativity to work, and one wonders whether these results will hold once quantum mechanics is taken into account in a theory of full quantum gravity.
The following references discuss general relativity.