  In this PhD thesis, we deal with problems related to nonlocal operators, in
particular to the fractional Laplacian and to some other types of fractional
derivatives (the Caputo and the Marchaud derivatives). We make an extensive
introduction to the fractional Laplacian, we present some related contemporary
research results and we add some original material. Indeed, we study the
potential theory of this operator, introduce a new proof of Schauder estimates
using the potential theory approach, we study a fractional elliptic problem in
$\mathbb{R}^n$ with convex nonlinearities and critical growth and we present a
stickiness property of nonlocal minimal surfaces for small values of the
fractional parameter. Also, we point out that the (nonlocal) character of the
fractional Laplacian gives rise to some surprising nonlocal effects. We prove
that other fractional operators have a similar behavior: in particular,
Caputo-stationary functions are dense in the space of smooth functions,
moreover, we introduce an extension operator for Marchaud-stationary functions.
