Aspects of compactifications and black holes in four-dimensional supergravity

In the 20th century, theoretical physics has seen the development of General Relativity and the Standard Model of elementary particles. These theories describe, with great precision, gravity and all known matter, respectively. However, it is not possible to unite them into one, single theory. We need such a theory of quantum gravitation to study phenomena where both theories are important, such as black holes and the Big Bang. One of the leading candidates for a theory of quantum gravitation is string theory. The low-energy limit of string theory is called supergravity. This thesis studies some of the open questions in string theory, in the setting of four-dimensional supergravity. We first classify the fully supersymmetric solutions of gauged N=2 supergravity with arbitrary gaugings in the vector- and hypermultiplet sectors. We present several examples of such solutions and connect some of them to vacuum solutions of flux compactifications in string theory. The next chapter searches for black hole solutions, where the scalars in the theory are gauged. Next, we show how one can embed known black hole solutions trough spontaneous symmetry breaking. When searching for new solutions, we find solutions only in cases when the metric contains ripples and the vector multiplet scalars become ghost-like. We then show how one can obtain four-dimensional supergravities by a Scherk-Schwarz reduction of eleven-dimensional supergravity, and study its equivalence with a compactification on a seven-dimensional space. Finally, we study general aspects of NS5-brane instantons in relation to the stabilization of the volume modulus in Calabi-Yau compactifications of type II strings with fluxes, and their orientifold versions.