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Time series models with time-dependent coefficients have appeared for a long time in the statistical, engineering and econometric literature of the last thirty years. Several approaches have been developed so that a review is welcome. We will distinguish spectral and time-domain, parametric and nonparametric, and univariate and multivariate approaches, but also the different asymptotic aspects. Absence of independence, stationarity and ergodicity implies that assumptions are still more delicate than in the general time series theory. Nevertheless, it is possible and practical. The author is more acquainted to the possible economic applications so this point of view is preferred. An illustration is given on a big dataset of US production series. ARIMA models are built automatically using TRAMO/SEATS. Then, the constant coefficients are replaced by linearly functions of time, the parameters of which being estimated by a quasi-maximum likelihood procedure. Finally, for each series, the constant and the time-dependent models are compared.