We define a nonlinear generalization of the singular value decomposition (SVD), which can be interpreted as a restricted SVD with Riemannian metrics in the column and row space. This so-called Riemannian SVD occurs in structured total least squares problems, for instance in the least squares approximation of a given matrix A by a rank deficient Hankel matrix B, which is an important problem in system identification and signal processing. Several algorithms to find the 'minimizing' singular triplet are suggested, both for the SVD and its nonlinear generalization. This paper reveals interesting connections between linear algebra (structured matrix problems), numerical analysis (algorithms), optimization theory, (differential) geometry and system theory (differential equations, stability, Lyapunov functions). We give some numerical examples and also point out some open problems.