Exact solutions in general relativity and alternative theories of gravity: mathematical and physical properties

In this thesis, we discuss several subjects connected with the framework of GR, in order to characterize astrophysical compact objects. The main purpose is to provide simple models describing gravitational fields generated by isolated compact bodies in stationary rotation with extremely simple internal structure, such as neutron stars. The main tools used for our analysis are exact solutions of Einstein fields equations, which have been approached in different ways. In particular, we use the formalism of junction conditions for finding new solutions of Einstein equations in presence of matter by matching metrics representing two shells of a compact body. With the same aim, we introduce the Newmann-Janis Algorithm, a solution generating technique which provides metrics of reduced symmetries from symmetric ones. Finally, an exact solution of Einstein's field equations, known as Einstein Static Universe is studied in the framework of Cosmology. Our purpose is to study the stability properties of this solution focusing on the intriguing possibility of finding static solutions in open cosmological models (k = -1). [edited by author]