  We report on recent progress in the study of nonlinear diffusion equations
involving nonlocal, long-range diffusion effects. Our main concern is the
so-called fractional porous medium equation, $\partial_t u
+(-\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual
porous medium flows, the fractional version has infinite speed of propagation
for all exponents $0<s<1$ and $m>0$; on the other hand, it also generates an
$L^1$-contraction semigroup which depends continuously on the exponent of
fractional differentiation and the exponent of the nonlinearity.
  After establishing the general existence and uniqueness theory, the main
properties are described: positivity, regularity, continuous dependence, a
priori estimates, Schwarz symmetrization, among others. Self-similar solutions
are constructed (fractional Barenblatt solutions) and they explain the
asymptotic behaviour of a large class of solutions. In the fast diffusion range
we study extinction in finite time and we find suitable special solutions. We
discuss KPP type propagation. We also examine some related equations that
extend the model and briefly comment on current work.
