  It is well-known that reliable and efficient domain truncation is crucial to
accurate numerical solution of most wave propagation problems. The perfectly
matched layer (PML) is a method which, when stable, can provide a domain
truncation scheme which is convergent with increasing layer width/damping. The
difficulties in using the PML are primarily associated with stability, which
can be present at the continuous level or be triggered by numerical
approximations. The mathematical and numerical analysis of the PML for
hyperbolic wave propagation problems has been an area of active research. It is
now possible to construct stable and high order accurate numerical wave solvers
by augmenting wave equations with the PML and approximating the equations using
summation-by-parts finite difference methods, continuous and discontinuous
Galerkin finite element methods. In this review we summarise the progress made,
from mathematical, numerical and practical perspectives, point out some open
problems and set the stage for future work. We also present numerical
experiments of model problems corroborating the theoretical analysis, and
numerical simulations of real-world wave propagation demonstrating impact.
Stable and parallel implementations of the PML in the high performance
computing software packages WaveQLab3D and ExaHyPE allow to sufficiently limit
the computational domain of seismological problems with only a few grid
points/elements around the computational boundaries where the PML is active,
thus saving as much as $96\%$ of the required computational resources for a
three space dimensional seismological benchmark problem.
