Many practical applications of statistical post-processing methods for ensemble weather forecasts require accurate modeling of spatial, temporal, and inter-variable dependencies. Over the past years, a variety of approaches has been proposed to address this need. We provide a comprehensive review and comparison of state-of-the-art methods for multivariate ensemble post-processing. We focus on generally applicable two-step approaches where ensemble predictions are first post-processed separately in each margin and multivariate dependencies are restored via copula functions in a second step. The comparisons are based on simulation studies tailored to mimic challenges occurring in practical applications and allow ready interpretation of the effects of different types of misspecifications in the mean, variance, and covariance structure of the ensemble forecasts on the performance of the post-processing methods. Overall, we find that the Schaake shuffle provides a compelling benchmark that is difficult to outperform, whereas the forecast quality of parametric copula approaches and variants of ensemble copula coupling strongly depend on the misspecifications at hand.