The tensor rank decomposition or canonical polyadic decomposition (CPD) is a generalization of a low-rank matrix factorization from matrices to higher-order tensors. In many applications, multi-dimensional data can be meaningfully approximated by a low-rank CPD. In this talk, I will describe a Riemannian optimization method for approximating a tensor by a low-rank CPD. This is a type of optimization method in which the domain is a smooth manifold, i.e. a curved geometric object. The presented method achieved up to two orders of magnitude improvements in execution time for challenging small-scale dense tensors when compared to state-of-the-art nonlinear least squares solvers.