  Many evolution problems in physics are described by partial differential
equations on an infinite domain; therefore, one is interested in the solutions
to such problems for a given initial dataset. A prominent example is the binary
black hole problem within Einstein's theory of gravitation, in which one
computes the gravitational radiation emitted from the inspiral of the two black
holes, merger and ringdown. Powerful mathematical tools can be used to
establish qualitative statements about the solutions, such as their existence,
uniqueness, continuous dependence on the initial data, or their asymptotic
behavior over large time scales. However, one is often interested in computing
the solution itself, and unless the partial differential equation is very
simple, or the initial data possesses a high degree of symmetry, this
computation requires approximation by numerical discretization. When solving
such discrete problems on a machine, one is faced with a finite limit to
computational resources, which leads to the replacement of the infinite
continuum domain with a finite computer grid. This, in turn, leads to a
discrete initial-boundary value problem. The hope is to recover, with high
accuracy, the exact solution in the limit where the grid spacing converges to
zero with the boundary being pushed to infinity.
  The goal of this article is to review some of the theory necessary to
understand the continuum and discrete initial-boundary value problems arising
from hyperbolic partial differential equations and to discuss its applications
to numerical relativity; in particular, we present well-posed initial and
initial-boundary value formulations of Einstein's equations, and we discuss
multi-domain high-order finite difference and spectral methods to solve them.
