  This review is focused on tests of Einstein's theory of General Relativity
with gravitational waves that are detectable by ground-based interferometers
and pulsar timing experiments. Einstein's theory has been greatly constrained
in the quasi-linear, quasi-stationary regime, where gravity is weak and
velocities are small. Gravitational waves will allow us to probe a
complimentary, yet previously unexplored regime: the non-linear and dynamical
strong-field regime. Such a regime is, for example, applicable to compact
binaries coalescing, where characteristic velocities can reach fifty percent
the speed of light and compactnesses can reach a half. This review begins with
the theoretical basis and the predicted gravitational wave observables of
modified gravity theories. The review continues with a brief description of the
detectors, including both gravitational wave interferometers and pulsar timing
arrays, leading to a discussion of the data analysis formalism that is
applicable for such tests. The review ends with a discussion of gravitational
wave tests for compact binary systems.
