We consider the fractional Laplace framework and provide models and theorems
related to nonlocal diffusion phenomena. Some applications are presented,
including: a simple probabilistic interpretation, water waves, crystal
dislocations, nonlocal phase transitions, nonlocal minimal surfaces and
Schr\"{o}dinger equations. Furthermore, an example of an $s$-harmonic function,
the harmonic extension and some insight on a fractional version of a classical
conjecture formulated by De Giorgi are presented. Although this book aims at
gathering some introductory material on the applications of the fractional
Laplacian, some proofs and results are original. Also, the work is self
contained, and the reader is invited to consult the rich bibliography for
further details, whenever a subject is of interest.