  In this thesis, we present various contributions to the study of free
boundary minimal surfaces. After introducing some basic tools and discussing
some delicate aspects related to the definition of Morse index when allowing
for a contact set, we divide the thesis in two parts. In the first part of this
dissertation, we study free boundary minimal surfaces with bounded Morse index
in a three-dimensional ambient manifold. More specifically, we present a
degeneration analysis of a sequence of such surfaces, proving that (up to
subsequence) they converge smoothly away from finitely many points and that,
around such `bad' points, we can at least `uniformly' control the topology and
the area of the surfaces in question. As a corollary, we obtain a complete
picture of the way different `complexity criteria' (in particular: topology,
area and Morse index) compare for free boundary minimal surfaces in ambient
manifolds with positive scalar curvature and mean convex boundary. In the
second part, we focus on an equivariant min-max scheme to prove the existence
of free boundary minimal surfaces with a prescribed topological type. The
principle is to choose a suitable group of isometries of the ambient manifold
in order to obtain exactly the topology we are looking for. We recall a proof
of the equivariant min-max theorem, and we also prove a bound on the Morse
index of the resulting surfaces. Finally, we apply this procedure to show the
existence of a new family of free boundary minimal surfaces with connected
boundary and arbitrary genus in the three-dimensional unit ball.
