This thesis concerns the optimization and application of low-rank methods, with a special focus on tensor trains (TTs). In particular, we develop methods for computing TT approximations of a given tensor in a variety of low-rank formats and we show how to solve the tensor completion problem for TTs using Riemannian methods. This is then applied to train a machine learning (ML) estimator based on discretized functions. We also study randomized methods for obtaining low-rank approximations of matrices and tensors. Finally, we consider how such randomized methods can be used to solve general linear matrix and tensor equations.