  Compact objects in general relativity approximately move along geodesics of
spacetime. It is shown that the corrections to geodesic motion due to spin
(dipole), quadrupole, and higher multipoles can be modeled by an extension of
the point mass action. The quadrupole contributions are discussed in detail for
astrophysical objects like neutron stars or black holes. Implications for
binaries are analyzed for a small mass ratio situation. There quadrupole
effects can encode information about the internal structure of the compact
object, e.g., in principle they allow a distinction between black holes and
neutron stars, and also different equations of state for the latter.
Furthermore, a connection between the relativistic oscillation modes of the
object and a dynamical quadrupole evolution is established.
