  The examination of parity symmetry in gravitational interactions has drawn
increasing attention. Although Einstein's General Relativity is
parity-conserved, numerous theories of parity-violating (PV) gravity in
different frameworks have recently been proposed for different motivations. In
this review, we briefly summarize the recent progress of these theories, and
focus on the observable effects of PV terms in the gravitational waves (GWs),
which are mainly reflected in the difference between the left-hand and
right-hand polarization modes. We are primarily concerned with the implications
of these theories for GWs generated by the compact binary coalescences and the
primordial GWs generated in the early Universe. The deviation of GW waveforms
and/or primordial power spectrum can always be quantified by the energy scale
of parity violation of the theory. Applying the current and future GW
observation from laser interferometers and cosmic microwave background
radiation, the current and potential constraints on the PV energy scales are
presented, which indicates that the parity symmetry of gravity can be tested in
high energy scale in this new era of gravitational waves.
