In this paper we provide a unified likelihood treatment of a wide class of non-Gaussian time series, semi-parametric and non-parametric regression, index models, spectral density estimation and panel data problems. The methods rely on the use of Markov chain Monte Carlo to carry out non-Gaussian simulation smoothing and Bayesian posterior analysis of parameters, and on importance sampling to estimate the likelihood function for classical inference. We exploit the structure of the models to ensure that our methods are highly efficient, delivering MCMC simulations with little autocorrelation and highly effective estimates of the likelihood.