This study investigates the problem of areostationary orbits around Mars in three-dimensional space.Areostationary orbits are expected to be used to establish a future telecommunication network for the exploration of Mars.However,no artificial satellites have been placed in these orbits thus far.The characteristics of the Martian gravity field are presented,and areostationary points and their linear stability are calculated.By taking linearized solutions in the planar case as the initial guesses and utilizing the Levenberg-Marquardt method,families of periodic orbits around areostationary points are shown to exist.Short-period orbits and long-period orbits are found around linearly stable areostationary points,but only short-period orbits are found around unstable areostationary points.Vertical periodic orbits around both linearly stable and unstable areostationary points are also examined.Satellites in these periodic orbits could depart from areostationary points by a few degrees in longitude,which would facilitate observation of the Martian topography.Based on the eigenvalues of the monodromy matrix,the evolution of the stability index of periodic orbits is determined.Finally,heteroclinic orbits connecting the two unstable areostationary points are found,providing the possibility for orbital transfer with minimal energy consumption.