We consider the problem of filtering and smoothing in state-space models, which include nonlinear and non-Gaussian models. We do not make any distributional assumptions about the processes involved. Our approach to these problems is based on the theory of estimating functions. Filter and smoother are obtained as solutions of estimating equations that are optimal in appropriate classes. We illustrate our procedures by simulation studies of a model where the observational variance depends on the state and a binomial logit model with a covariate. In non-Gaussian cases, procedures based on estimating equations often perform considerably better than the existing semiparametric procedures.