Introduction
1 A probabilistic motivation.-1.1 The random walk with arbitrarily long jumps
1.2 A payoff model.-2 An introduction to the fractional Laplacian.-2.1 Preliminary notions
2.2 Fractional Sobolev Inequality and Generalized Coarea Formula
2.3 Maximum Principle and Harnack Inequality
2.4 An s-harmonic function
2.5 All functions are locally s-harmonic up to a small error
2.6 A function with constant fractional Laplacian on the ball
3 Extension problems
3.1 Water wave model
3.2 Crystal dislocation
3.3 An approach to the extension problem via the Fourier transform
4 Nonlocal phase transitions
4.1 The fractional Allen-Cahn equation
4.2 A nonlocal version of a conjecture by De Giorgi
5 Nonlocal minimal surfaces
5.1 Graphs and s-minimal surfaces
5.2 Non-existence of singular cones in dimension 2 5.3 Boundary regularity
6 A nonlocal nonlinear stationary Schrödinger type equation
6.1 From the nonlocal Uncertainty Principle to a fractional weighted inequality
Alternative proofs of some results
A.1 Another proof of Theorem A.2 Another proof of Lemma 2.3
References.