The new era of gravitational wave astronomy truly began on September 14, 2015, with the detection of GW150914, the sensational first direct observation of gravitational waves from the inspiral and merger of two black holes by the two Advanced LIGO detectors. In the subsequent first three observing runs of the LIGO-Virgo network, gravitational waves from $\sim 50$ compact binary mergers have been announced, with more results to come. The events have mostly been produced by binary black holes, but two binary neutron star mergers have been observed thus far, as well as the mergers of two neutron star–black hole systems. Furthermore, gravitational waves emitted by core-collapse supernovae, pulsars, and the stochastic gravitational wave background are within the LIGO-Virgo-KAGRA sensitivity band and are likely to be observed in future observation runs. Beyond signal detection, a major challenge has been the development of statistical and computational methodology for estimating the physical waveform parameters and quantifying their uncertainties in order to accurately characterize the emitting system. These methods depend on the sources of the gravitational waves and the gravitational waveform model that is used. This review examines the main waveform models and parameter estimation methods used to extract physical parameters from gravitational wave signals detected to date by LIGO and Virgo and from those expected to be observed in the future, which will include KAGRA, and how these methods interface with various aspects of LIGO-Virgo-KAGRA science. Also presented are the statistical methods used by LIGO and Virgo to estimate detector noise, test general relativity, and draw conclusions about the rates of compact binary mergers in the Universe. Furthermore, a summary of major publicly available gravitational wave parameter estimation software packages is given.