We develop methods for performing filtering and smoothing computations in non-linear non-Gaussian state space models. The methods rely on a particle representation of the filtering distributions, and their evolution through time using sequential importance sampling and resampling ideas. In particular, novel techniques are presented for generation of sample realisations of historical state sequences. This is carried out in a forward-filtering backward-smoothing precedure which can be viewed as the non-linear, non-Gaussian counterpart of standard Kalman filter-based simulation smoothers in the linear Gaussian case. The methods are tested for a standard non-linear time series model and a substantial application is developed for processing of speech signals represented by a time-varying autoregression and parameterised in terms of its time-varying partial correlation coefficients

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