In this talk several issues of the fractional nonlinear Schrödinger equation are analysed. Some properties of the equation enable the use of Concentration-Compactness theory to prove the existence of solitary wave solutions with algebraic decay. Then a numerical approach introduces a full discretization of the periodic initial-value problem and derives error estimates. From the numerical generation of the solitary wave profiles and the fully discrete scheme, a computational study of the dynamics of the solitary waves is developed. This is a joint work with Nuria Reguera.