The detection of gravitational waves by the LIGO-Virgo-KAGRA collaboration has ushered in a new era of observational astronomy, emphasizing the need for rapid and detailed parameter estimation and population-level analyses. Traditional Bayesian inference methods, particularly Markov chain Monte Carlo, face significant computational challenges when dealing with the high-dimensional parameter spaces and complex noise characteristics inherent in gravitational wave data. This review examines the emerging role of simulation-based inference methods in gravitational wave astronomy, with a focus on approaches that leverage machine-learning techniques such as normalizing flows and neural posterior estimation. We provide a comprehensive overview of the theoretical foundations underlying various simulation-based inference methods, including neural posterior estimation, neural ratio estimation, neural likelihood estimation, flow matching, and consistency models. We explore the applications of these methods across diverse gravitational wave data processing scenarios, from single-source parameter estimation and overlapping signal analysis to testing general relativity and conducting population studies. Although these techniques demonstrate speed improvements over traditional methods in controlled studies, their model-dependent nature and sensitivity to prior assumptions are barriers to their widespread adoption. Their accuracy, which is similar to that of conventional methods, requires further validation across broader parameter spaces and noise conditions.