Birkhoff periodic orbits in celestial mechanics, especially the commensurabilities between the revolutional and the rotational period, the so-called spin-orbit resonances, are studied. The general motivation for such investigations comes from the abundance of resonant relations in the solar system. In most of the cases, the evolved satellites of the solar system have a 1:1 or synchronous spin-orbit resonance, that is they always point the same face to the host planet (e.g. Moon-Earth). The only exception is the Mercury-Sun system in approximately 3:2 resonance. The authors introduce a mathematical model describing an approximation of the spin orbit problem which in particular reduces to the study of a Hamiltonian equation for the librational angle including a perturbative parameter for the equatorial oblateness of the satellite and a smooth periodic function depending on the eccentricity of the Keplerian orbit. A general method based on a quantitative version of the classical implicit function theorem to construct Birkhoff periodic orbits (which have as their frequency the resonance ratio) is presented. Some results for the 1:1, 3:2, 2:1 resonances in the Moon-Earth and Sun-Mercury systems are largely discussed and compared with observed data on the libration in longitude given in the Astronomical Almanac. Furthermore considerations on the stability of the constructed periodic orbits are made. Finally, the authors show that Lyapunov stability can be obtained by proving the existence of librational KAM invariant surfaces trapping the periodic orbits and that the Siegel-Moser condition for the existence of librational invariant surfaces are satisfied.