We propose a new generic and highly efficient Accelerated Gaussian Importance Sampler (AGIS) for the numerical evaluation of (very) high-dimensional density functions. A specific case of interest to us is the evaluation of likelihood functions for a broad class of dynamic latent variable models. The feasibility of our method is strikingly illustrated by means of an application to a first-order dynamic stochastic volatility model for daily stock returns, whose likelihood for an actual sample of size 2022(!) is evaluated with high numerical accuracy by means of 10,000 Monte Carlo replications. The estimated model parsimoniously dominates ARCH and GARCH alternatives, one of which includes twelve lags. Copyright 1993 by John Wiley & Sons, Ltd.