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Purpose: The finite element method (FEM) is the preferred method to simulate phenomena in anatomical structures. However, purely FEM-based mechanical simulations require considerable time, limiting their use in clinical applications that require real-time responses, such as haptics simulators. Machine learning (ML) approaches have been proposed to help with the reduction of the required time. The present paper reviews cases where ML could help to generate faster simulations, without considerably affecting the performance results., Methods: This review details the ML approaches used, considering the anatomical structures involved, the data collection strategies, the selected ML algorithms, with corresponding features, the metrics used for validation, and the resulting time gains., Results: A total of 41 references were found. ML algorithms are mainly trained with FEM-based simulations in 32 publications. The preferred ML approach is neural networks, including deep learning in 35 publications. Tissue deformation is simulated in 18 applications, but other features are also considered. The average distance error and mean squared error are the most frequently used performance metrics, in 14 and 17 publications, respectively. The time gains were considerable, going from hours or minutes for purely FEM-based simulations to milliseconds, when using ML., Conclusions: ML algorithms can be used to accelerate FEM-based biomechanical simulations of anatomical structures, possibly reaching real-time responses. Fast and real-time simulations of anatomical structures, generated with ML algorithms, can help to reduce the time required by FEM-based simulations and accelerate their adoption in the clinical practice., (© 2020 American Association of Physicists in Medicine.)