This paper provides a comprehensive survey of deformable medical image registration methods, emphasizing recent advances and their application to medical imaging. Deformable registration is a fundamental task in medical image processing, with key applications including multi-modality fusion, longitudinal studies, and population modeling. The paper systematically reviews the three main components of registration algorithms: deformation models, matching criteria, and optimization methods. It discusses various deformation models, including elastic body models, viscous fluid flow models, diffusion models, curvature registration, and diffeomorphic flows. Each model is analyzed for its ability to capture different types of deformations and its computational efficiency. The paper also addresses the challenges of ill-posedness in registration problems and the importance of inverse consistency, symmetry, and topology preservation in ensuring accurate and reliable results. Additionally, it covers interpolation-based deformation models, such as radial basis functions and thin-plate splines, which are widely used in medical image registration due to their flexibility and ability to handle local deformations. The paper highlights the importance of regularization in ensuring smooth and bijective transformations, and discusses various optimization techniques for solving the registration problem. It also addresses the application of registration methods to different types of medical data, including scalar, vector, and tensor-valued images, as well as to different geometries. The paper concludes by emphasizing the need for further research to improve the efficiency and accuracy of deformable registration methods, particularly in the context of clinical applications.This paper provides a comprehensive survey of deformable medical image registration methods, emphasizing recent advances and their application to medical imaging. Deformable registration is a fundamental task in medical image processing, with key applications including multi-modality fusion, longitudinal studies, and population modeling. The paper systematically reviews the three main components of registration algorithms: deformation models, matching criteria, and optimization methods. It discusses various deformation models, including elastic body models, viscous fluid flow models, diffusion models, curvature registration, and diffeomorphic flows. Each model is analyzed for its ability to capture different types of deformations and its computational efficiency. The paper also addresses the challenges of ill-posedness in registration problems and the importance of inverse consistency, symmetry, and topology preservation in ensuring accurate and reliable results. Additionally, it covers interpolation-based deformation models, such as radial basis functions and thin-plate splines, which are widely used in medical image registration due to their flexibility and ability to handle local deformations. The paper highlights the importance of regularization in ensuring smooth and bijective transformations, and discusses various optimization techniques for solving the registration problem. It also addresses the application of registration methods to different types of medical data, including scalar, vector, and tensor-valued images, as well as to different geometries. The paper concludes by emphasizing the need for further research to improve the efficiency and accuracy of deformable registration methods, particularly in the context of clinical applications.