The resonant structure of the restricted three-body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2:1, 3:1 and 4:1 mean motion resonance of the asteroid with Jupiter. In each case we start with the unperturbed two-body system ``Sun-asteroid'', and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter.