  We describe the mathematical theory of diffusion and heat transport with a
view to including some of the main directions of recent research. The linear
heat equation is the basic mathematical model that has been thoroughly studied
in the last two centuries. It was followed by the theory of parabolic equations
of different types. In a parallel development, the theory of stochastic
differential equations gives a foundation to the probabilistic study of
diffusion. Nonlinear diffusion equations have played an important role not only
in theory but also in physics and engineering, and we focus on a relevant
aspect, the existence and propagation of free boundaries. We use the porous
medium and fast diffusion equations as case examples. A large part of the paper
is devoted to diffusion driven by fractional Laplacian operators and other
nonlocal integro-differential operators representing nonlocal, long-range
diffusion effects. Three main models are examined (one linear, two nonlinear),
and we report on recent progress in which the author is involved.
