  Applying the method of analytical continuation of periodic orbits, we study
quasi-satellite motion in the framework of the three-body problem. In the
simplest, yet not trivial model, namely the planar circular restricted problem,
it is known that quasi-satellite motion is associated with a family of periodic
solutions, called family $f$, which consists of 1:1 resonant retrograde orbits.
In our study, we determine the critical orbits of family $f$ that are continued
both in the elliptic and in the spatial model and compute the corresponding
families that are generated and consist the backbone of the quasi-satellite
regime in the restricted model. Then, we show the continuation of these
families in the general three-body problem, we verify and explain previous
computations and show the existence of a new family of spatial orbits. The
linear stability of periodic orbits is also studied. Stable periodic orbits
unravel regimes of regular motion in phase space where 1:1 resonant angles
librate. Such regimes, which exist even for high eccentricities and
inclinations, may consist dynamical regions where long-lived asteroids or
co-orbital exoplanets can be found.
