  Our investigation is motivated by the recent discovery of asteroids orbiting
the Sun and simultaneously staying near one of the Solar System planets for a
long time. This regime of motion is usually called the quasi-satellite regime,
since even at the times of the closest approaches the distance between the
asteroid and the planet is significantly larger than the region of space (the
Hill's sphere) in which the planet can hold its satellites. We explore the
properties of the quasi-satellite regimes in the context of the spatial
restricted circular three-body problem "Sun-planet-asteroid". Via double
numerical averaging, we construct evolutionary equations which describe the
long-term behaviour of the orbital elements of an asteroid. Special attention
is paid to possible transitions between the motion in a quasi-satellite orbit
and the one in another type of orbits available in the 1:1 resonance. A rough
classification of the corresponding evolutionary paths is given for an
asteroid's motion with a sufficiently small eccentricity and inclination.
