  The detections of gravitational-wave (GW) signals from compact binary
coalescence by ground-based detectors have opened up the era of GW astronomy.
These observations provide opportunities to test Einstein's general theory of
relativity at the strong-field regime. Here we give a brief overview of the
various GW-based tests of General Relativity (GR) performed by the LIGO-Virgo
collaboration on the detected GW events to date. After providing details for
the tests performed in four categories, we discuss the prospects for each test
in the context of future GW detectors. The four categories of tests include the
consistency tests, parametrized tests for GW generation and propagation, tests
for the merger remnant properties, and GW polarization tests.
