  In this thesis we investigate how the nonlocalities affect the study of
different PDEs coming from physics, and we analyze these equations under almost
optimal assumptions of the nonlinearity. In particular, we focus on the
fractional Laplacian operator and on sources involving convolution with the
Riesz potential, as well as on the interaction of the two, and we aim to do it
through variational and topological methods.
  We examine both quantitative and qualitative aspects, proving multiplicity of
solutions for nonlocal nonlinear problems with free or prescribed mass, showing
regularity, positivity, symmetry and sharp asymptotic decay of ground states,
and exploring the influence of the topology of a potential well in presence of
concentration phenomena. On the nonlinearities we consider general assumptions
which avoid monotonicity and homogeneity: this generality obstructs the use of
classical variational tools and forces the implementation of new ideas.
  Throughout the thesis we develop some new tools: among them, a Lagrangian
formulation modeled on Pohozaev mountains is used for the existence of
normalized solutions, annuli-shaped multidimensional paths are built for
genus-based multiplicity results, a fractional chain rule is proved to treat
concave powers, and a fractional center of mass is defined to detect
semiclassical standing waves. We believe that these tools could be used to face
problems in different frameworks as well.
