The key player is the problem: minimize f(x), where x lives on a smooth manifold M, and f is a smooth cost function on M. Applications abound in scientific computing, signal processing, computer vision, machine learning and statistics. Manifolds arise in optimization as a result of constraints (e.g., low-rank, orthogonality) and as a result of symmetry (quotient spaces). By endowing the manifold with a Riemannian structure, we obtain meaningful notions of gradient and Hessian on the manifold. This enables us to generalize standard algorithms such as gradient descent and trust-regions. The course mixes mathematical analysis and coding.