  Newtonian point mass binaries can be brought into arbitrarily close circular
orbits. Neutron stars and black holes, however, are extended, relativistic
objects. Both finite size and relativistic effects make very close orbits
unstable, so that there exists an innermost stable circular orbit (ISCO). We
illustrate the physics of the ISCO in a simple model problem, and review
different techniques which have been employed to locate the ISCO in black hole
and neutron star binaries. We discuss different assumptions and approximations,
and speculate on how differences in the results may be explained and resolved.
