Manopt is a MATLAB toolbox for optimization on manifolds, designed to simplify the use of Riemannian optimization algorithms. It is aimed at practitioners outside the field of nonlinear optimization. The toolbox provides a user-friendly interface for experimenting with state-of-the-art algorithms, with documentation and code available at www.manopt.org. Optimization on manifolds involves solving problems where the search space is a smooth manifold, allowing for efficient numerical algorithms. This approach is particularly useful for problems with rank and orthogonality constraints, which are common in machine learning applications such as low-rank matrix completion, sensor network localization, and metric learning. Manopt supports various manifolds, including the Stiefel manifold, Grassmann manifold, special orthogonal group, fixed-rank matrices, and fixed-rank spectrahedrons. These manifolds allow for the optimization of functions with specific constraints, such as rank or orthogonality. The toolbox includes a range of solvers for Riemannian optimization, including trust-region methods, conjugate gradients, and steepest descent. It also provides tools for checking the correctness of gradients and Hessians, and for caching intermediate results to improve performance. An example application is the max-cut problem, which can be solved using Riemannian optimization on the fixed-rank elliptope. The Manopt toolbox provides code for this example, demonstrating how to optimize over the manifold and obtain a solution. The toolbox is supported by a range of references and is designed to be flexible and extensible, allowing for the addition of new manifolds and solvers as needed. It is a valuable resource for researchers and practitioners working in the field of nonlinear optimization and related areas.Manopt is a MATLAB toolbox for optimization on manifolds, designed to simplify the use of Riemannian optimization algorithms. It is aimed at practitioners outside the field of nonlinear optimization. The toolbox provides a user-friendly interface for experimenting with state-of-the-art algorithms, with documentation and code available at www.manopt.org. Optimization on manifolds involves solving problems where the search space is a smooth manifold, allowing for efficient numerical algorithms. This approach is particularly useful for problems with rank and orthogonality constraints, which are common in machine learning applications such as low-rank matrix completion, sensor network localization, and metric learning. Manopt supports various manifolds, including the Stiefel manifold, Grassmann manifold, special orthogonal group, fixed-rank matrices, and fixed-rank spectrahedrons. These manifolds allow for the optimization of functions with specific constraints, such as rank or orthogonality. The toolbox includes a range of solvers for Riemannian optimization, including trust-region methods, conjugate gradients, and steepest descent. It also provides tools for checking the correctness of gradients and Hessians, and for caching intermediate results to improve performance. An example application is the max-cut problem, which can be solved using Riemannian optimization on the fixed-rank elliptope. The Manopt toolbox provides code for this example, demonstrating how to optimize over the manifold and obtain a solution. The toolbox is supported by a range of references and is designed to be flexible and extensible, allowing for the addition of new manifolds and solvers as needed. It is a valuable resource for researchers and practitioners working in the field of nonlinear optimization and related areas.