Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance. "The book under review is a result of a series of lectures given in various places throughout the world. It gives an introduction to the analysis of nonlocal operators, most notably the fractional Laplacian. ? the book does a great job of introducing the topic of nonlocal analysis for every newcomer in the field. It provides a good starting point for doing research and therefore is highly recommended." (Lukasz Plociniczak, Mathematical Reviews, March, 2017)