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The use of perturbation techniques in General Relativity dates back to the very beginnings, when the weak nature of gravity and the slow motion of planets in the solar system were exploited to build approximation methods. Since then these methods have been refined, and new methods have been invented to solve new problems.

For instance, the post-Newtonian and post-Minkowskian formalisms aim to find approximate solutions to the Einstein field equations when the gravitational field is weak and the motion of bodies is slow. These formalisms have been exploited to calculate the metric of an N-body system, to obtain the equations of motion satisfied by these bodies, and to extract the gravitational waves generated by these motions. Black-hole perturbation theory describes an isolated black hole perturbed slightly by nearby objects, and it has produced a host of interesting phenomena such as the quasinormal ringdown of black holes, the gravitational waves emitted by a body in very rapid motion in the black hole's very strong field, and the gravitational self-force acting on this body. Recently, the powerful body of techniques known as effective field theory, first developed in the context of quantum field theory, have been imported to General Relativity and have profitably informed the traditional perturbative approaches. While these techniques are all approximations that rely on the existence of a smallness parameter (such as the ratio of velocities to the speed of light in the case of post-Newtonian theory, or the mass ratio in the case of a black hole perturbed by a small body), a different kind of approximation is delivered when the Einstein field equations are discretized and solved on supercomputers.

The recent breakthroughs of numerical relativity have allowed us to understand the rich dynamics of the gravitational field during black hole collisions, the instability of higher-dimensional black strings, and is now shedding light on the interaction between neutron stars, their accretion disks, and their magnetic fields. Numerical relativity is also increasingly informing the perturbative methods, for instance through comparisons with post-Newtonian approximations, and through the numerical calculation of new self-force results.

These methods are all being applied to improve our understanding of Einstein's equations and their practical applications, but they rely strongly on either known or believed fundamental properties of the underlying mathematical structure of the theory. For instance, when black-hole perturbation theory is applied to the stability of black holes, it is assumed that the underlying system of partial differential equations admits a well-posed initial-value problem. As another example, the convergence of the post-Newtonian sequence of approximations is still an open problem awaiting mathematical attention. It is clear that a close dialogue between the physical and mathematical communities is important, as this will help further not only the individual research agendas but also cross-pollinate ideas and problems through dialogue, discussions, and collaborations.