The paper introduces Manopt, a MATLAB toolbox designed for optimization on manifolds, which is particularly useful for problems with rank and orthogonality constraints. These constraints are common in machine learning applications such as low-rank matrix completion, sensor network localization, and camera network registration. The toolbox aims to simplify the use of advanced Riemannian optimization algorithms, making them accessible to practitioners outside the field of nonlinear optimization. The authors highlight the importance of leveraging the smooth geometry of the search space to design efficient numerical algorithms. They provide examples of manifolds and optimization problems, such as the oblique manifold and the fixed-rank ellipotope, and discuss the theoretical foundations of Riemannian optimization, including the definition of gradients and Hessians, and the use of retractions. Manopt's architecture is designed to be user-friendly, with a separation of manifolds, solvers, and problem descriptions. The toolbox includes various solvers for Riemannian minimization, such as Riemannian trust-regions and conjugate-gradients, and supports common stopping criteria. The paper also provides an example of solving the max-cut problem using Manopt, demonstrating how to define the problem structure, compute gradients and Hessians, and apply optimization algorithms. The authors acknowledge the support from the FNRS, the Belgian Network DYSCO, and the Belgian FRFC.The paper introduces Manopt, a MATLAB toolbox designed for optimization on manifolds, which is particularly useful for problems with rank and orthogonality constraints. These constraints are common in machine learning applications such as low-rank matrix completion, sensor network localization, and camera network registration. The toolbox aims to simplify the use of advanced Riemannian optimization algorithms, making them accessible to practitioners outside the field of nonlinear optimization. The authors highlight the importance of leveraging the smooth geometry of the search space to design efficient numerical algorithms. They provide examples of manifolds and optimization problems, such as the oblique manifold and the fixed-rank ellipotope, and discuss the theoretical foundations of Riemannian optimization, including the definition of gradients and Hessians, and the use of retractions. Manopt's architecture is designed to be user-friendly, with a separation of manifolds, solvers, and problem descriptions. The toolbox includes various solvers for Riemannian minimization, such as Riemannian trust-regions and conjugate-gradients, and supports common stopping criteria. The paper also provides an example of solving the max-cut problem using Manopt, demonstrating how to define the problem structure, compute gradients and Hessians, and apply optimization algorithms. The authors acknowledge the support from the FNRS, the Belgian Network DYSCO, and the Belgian FRFC.