  We review state-of-the-art theory and experiment of the motion of cold and
ultracold atoms coupled to the radiation field within a high-finesse optical
resonator in the dispersive regime of the atom-field interaction with small
internal excitation. The optical dipole force on the atoms together with the
back-action of atomic motion onto the light field gives rise to a complex
nonlinear coupled dynamics. As the resonator constitutes an open driven and
damped system, the dynamics is non-conservative and in general enables cooling
and confining the motion of polarizable particles. In addition, the emitted
cavity field allows for real-time monitoring of the particle's position with
minimal perturbation up to sub-wavelength accuracy. For many-body systems, the
resonator field mediates controllable long-range atom-atom interactions, which
set the stage for collective phenomena. Besides correlated motion of distant
particles, one finds critical behavior and non-equilibrium phase transitions
between states of different atomic order in conjunction with superradiant light
scattering. Quantum degenerate gases inside optical resonators can be used to
emulate opto-mechanics as well as novel quantum phases like supersolids and
spin glasses. Non-equilibrium quantum phase transitions, as predicted by e.g.
the Dicke Hamiltonian, can be controlled and explored in real-time via
monitoring the cavity field. In combination with optical lattices, the cavity
field can be utilized for non-destructive probing Hubbard physics and tailoring
long-range interactions for ultracold quantum systems.
