It is shown how structured and weighted total least squares and L₂ approximation problems lead to a “nonlinear” generalized singular value decomposition. An inverse iteration scheme to find a (local) minimum is proposed. The emphasis of the paper is not on the convergence analysis of the algorithm; rather the purpose is to illustrate its use in numerous applications in systems and control, including total least squares with relative errors and/or fixed elements, inverse singular value problems, an errors-in-variables variant of the Kalman filter, impulse response realization from noisy data, H₂ model reduction, H₂ system identification, and calculating the largest stability radius of uncertain linear systems. Several numerical examples are given.