The paper provides a comprehensive overview of deformable image registration methods, emphasizing recent advances and their applications in medical image analysis. Deformable registration is a fundamental task in medical image processing, used for multi-modality fusion, longitudinal studies, and population modeling. The process involves establishing spatial correspondences between different image acquisitions, typically involving a source (moving) image and a target (fixed) image. The goal is to estimate an optimal transformation that minimizes an objective function combining a matching criterion and a regularization term. The paper is structured into three main components: deformation models, matching criteria, and optimization methods. Deformation models are categorized into geometric transformations derived from physical models, interpolation theory, and knowledge-based models. Geometric transformations include elastic body models, viscous fluid flow models, diffusion models, curvature registration, and flows of diffeomorphisms. Interpolation theory-based models use radial basis functions, thin-plate splines, and free-form deformations. Knowledge-based models incorporate specific prior information about the sought deformation. The matching criteria section discusses various distance measures and similarity metrics used to quantify the alignment between images. Regularization terms aim to enforce properties such as inverse consistency, symmetry, topology preservation, and diffeomorphism. The optimization methods section covers techniques for solving the registration problem, including gradient descent, variational approaches, and numerical schemes. The paper highlights the challenges and advancements in deformable registration, emphasizing the importance of choosing appropriate deformation models and optimization strategies to achieve accurate and efficient registration results. It also discusses the clinical impact of these methods and their applications in medical image analysis.The paper provides a comprehensive overview of deformable image registration methods, emphasizing recent advances and their applications in medical image analysis. Deformable registration is a fundamental task in medical image processing, used for multi-modality fusion, longitudinal studies, and population modeling. The process involves establishing spatial correspondences between different image acquisitions, typically involving a source (moving) image and a target (fixed) image. The goal is to estimate an optimal transformation that minimizes an objective function combining a matching criterion and a regularization term. The paper is structured into three main components: deformation models, matching criteria, and optimization methods. Deformation models are categorized into geometric transformations derived from physical models, interpolation theory, and knowledge-based models. Geometric transformations include elastic body models, viscous fluid flow models, diffusion models, curvature registration, and flows of diffeomorphisms. Interpolation theory-based models use radial basis functions, thin-plate splines, and free-form deformations. Knowledge-based models incorporate specific prior information about the sought deformation. The matching criteria section discusses various distance measures and similarity metrics used to quantify the alignment between images. Regularization terms aim to enforce properties such as inverse consistency, symmetry, topology preservation, and diffeomorphism. The optimization methods section covers techniques for solving the registration problem, including gradient descent, variational approaches, and numerical schemes. The paper highlights the challenges and advancements in deformable registration, emphasizing the importance of choosing appropriate deformation models and optimization strategies to achieve accurate and efficient registration results. It also discusses the clinical impact of these methods and their applications in medical image analysis.