Critical decisions frequently rely on high-dimensional output from complex computer simulation models that show intricate cross-variable, spatial and temporal dependence structures, with weather and climate predictions being key examples. There is a strongly increasing recognition of the need for uncertainty quantification in such settings, for which we propose and review a general multi-stage procedure called ensemble copula coupling (ECC), proceeding as follows: ¶ 1. Generate a raw ensemble, consisting of multiple runs of the computer model that differ in the inputs or model parameters in suitable ways. ¶ 2. Apply statistical postprocessing techniques, such as Bayesian model averaging or nonhomogeneous regression, to correct for systematic errors in the raw ensemble, to obtain calibrated and sharp predictive distributions for each univariate output variable individually. ¶ 3. Draw a sample from each postprocessed predictive distribution. ¶ 4. Rearrange the sampled values in the rank order structure of the raw ensemble to obtain the ECC postprocessed ensemble. ¶ The use of ensembles and statistical postprocessing have become routine in weather forecasting over the past decade. We show that seemingly unrelated, recent advances can be interpreted, fused and consolidated within the framework of ECC, the common thread being the adoption of the empirical copula of the raw ensemble. Depending on the use of Quantiles, Random draws or Transformations at the sampling stage, we distinguish the ECC-Q, ECC-R and ECC-T variants, respectively. We also describe relations to the Schaake shuffle and extant copula-based techniques. In a case study, the ECC approach is applied to predictions of temperature, pressure, precipitation and wind over Germany, based on the 50-member European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble.