  Observations of compact objects in the electromagnetic spectrum and the
detection of gravitational waves from them can lead to quantitative tests of
the theory of general relativity in the strong-field regime following two very
different approaches. In the first approach, the general relativistic field
equations are modified at a fundamental level and the magnitudes of the
potential deviations are constrained by comparison with observations. In the
second approach, the exterior spacetimes of compact objects are parametrized in
a phenomenological way, the various parameters are measured observationally,
and the results are finally compared against the general relativistic
predictions. In this article, I discuss the current status of both approaches,
focusing on the lessons learned from a large number of recent investigations.
