This paper provides a comprehensive survey of Approximate Bayesian Computational (ABC) methods, also known as likelihood-free methods. These methods have gained prominence in recent years as a solution to intractable likelihood problems, particularly in genetics and other applications. The authors discuss the origins and justifications of ABC, its calibration, recent improvements, post-processing techniques, and its application to model choice. They highlight the challenges of choosing summary statistics and the impact of tolerance levels on the accuracy of the ABC approximation. The paper also explores sequential Monte Carlo (SMC) methods, importance sampling, and non-parametric approaches to enhance the efficiency and accuracy of ABC. Additionally, it addresses the use of ABC for model comparison, including the limitations and potential of likelihood-free methods in this context. The authors provide practical examples and illustrate the methods with simulations, emphasizing the importance of careful calibration and the choice of appropriate summary statistics.This paper provides a comprehensive survey of Approximate Bayesian Computational (ABC) methods, also known as likelihood-free methods. These methods have gained prominence in recent years as a solution to intractable likelihood problems, particularly in genetics and other applications. The authors discuss the origins and justifications of ABC, its calibration, recent improvements, post-processing techniques, and its application to model choice. They highlight the challenges of choosing summary statistics and the impact of tolerance levels on the accuracy of the ABC approximation. The paper also explores sequential Monte Carlo (SMC) methods, importance sampling, and non-parametric approaches to enhance the efficiency and accuracy of ABC. Additionally, it addresses the use of ABC for model comparison, including the limitations and potential of likelihood-free methods in this context. The authors provide practical examples and illustrate the methods with simulations, emphasizing the importance of careful calibration and the choice of appropriate summary statistics.