Gromov hyperbolicity grasps the essence of both negatively curved spaces and discrete spaces. The hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it; hence, characterizing hyperbolic graphs is a main problem in the theory of hyperbolicity. Since this is a very ambitious goal, a more achievable problem is to characterize hyperbolic graphs in particular classes of graphs. The main result in this paper is a characterization of the hyperbolicity of periodic graphs.