In recent years, there has been a surge of activity focused on the use of so-called fractional diffusion operators to replace the standard Laplace operator, with the aim of further extending the theory by taking into account the presence of so-called long range interactions. These nonlocal operators can be found in an extend range of fields, such as mathematical finance, physics, image processing and geometry. The aim of this work is to present an introduction to the fractional Laplacian operator and study some specific problems not yet appeared in the literature, related to periodic solutions.