  We discuss some results related to a phase transition model in which the
potential energy induced by a double-well function is balanced by a fractional
elastic energy. In particular, we present asymptotic results (such as
$\Gamma$-convergence, energy bounds and density estimates for level sets),
flatness and rigidity results, and the construction of planelike minimizers in
periodic media.
  Finally, we consider a nonlocal equation with a multiwell potential,
motivated by models arising in crystal dislocations, and we construct orbits
exhibiting symbolic dynamics, inspired by some classical results by Paul
Rabinowitz.
