  This doctoral thesis is devoted to the analysis of some minimization problems
that involve nonlocal functionals. We are mainly concerned with the
$s$-fractional perimeter and its minimizers, the $s$-minimal sets. We
investigate the behavior of sets having (locally) finite fractional perimeter
and we establish existence and compactness results for (locally) $s$-minimal
sets. We study the $s$-minimal sets in highly nonlocal regimes, that correspond
to small values of the fractional parameter $s$. We introduce a functional
framework for studying those $s$-minimal sets that can be globally written as
subgraphs. In particular, we prove existence and uniqueness results for
minimizers of a fractional version of the classical area functional and we show
the equivalence between minimizers and various notions of solution of the
fractional mean curvature equation. We also prove a flatness result for entire
nonlocal minimal graphs having some partial derivatives bounded from either
above or below. Moreover, we consider a free boundary problem, which consists
in the minimization of a functional defined as the sum of a nonlocal energy,
plus the classical perimeter. Concerning this problem, we prove uniform energy
estimates and we study the blow-up sequence of a minimizer---in particular
establishing a Weiss-type monotonicity formula.
