IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.