Artificial intelligence and mechanical engineering are two mature fields of science that intersect more and more often. Computer-aided mechanical analysis tools, often multibody system software, are very versatile and have revolutionalized many industries. However, as shown by the literature presented in this review, combining the advantages of multibody system dynamics and machine learning creates new and exciting possibilities. For example, the multibody method can assist machine learning by providing synthetic data, while machine learning can provide fast and accurate subsystem models. The intersection of both approaches results in surrogate and hybrid modeling techniques, advanced control algorithms, and optimal design applications. A notable example is the development of autonomous systems for vehicles, robots, and mobile machinery. In our review we have found nontrivial, innovative, and even surprising applications of machine learning and multibody dynamics. This review focuses on applying neural networks, mainly deep learning, in connection with the multibody system method. Over one hundred and fifty papers are covered, and three main research areas are identified and introduced: data-driven modeling, model-based control and estimation, and data-driven control. The paper starts with a primer on machine learning and concludes with future research directions. The main goal is to provide a comprehensive and up-to-date review of existing literature to inspire further research.