AbstractRegression models describing the joint distribution of multivariate responses conditional on covariate information have become an important aspect of contemporary regression analysis. However, a limitation of such models are the rather simplistic assumptions often made, for example, a constant dependence structure not varying with covariates or the restriction to linear dependence between the responses. We propose a general framework for multivariate conditional transformation models that overcomes these limitations and describes the entire distribution in a tractable and interpretable yet flexible way conditional on nonlinear effects of covariates. The framework can be embedded into likelihood‐based inference, including results on asymptotic normality, and allows the dependence structure to vary with covariates. In addition, it scales well‐beyond bivariate response situations, which were the main focus of most earlier investigations. We illustrate the benefits in a trivariate analysis of childhood undernutrition and demonstrate empirically that complex truly multivariate data‐generating processes can be inferred from observations.