While statistical modeling of distributional data has gained increased attention, the case of multivariate distributions has been somewhat neglected despite its relevance in various applications. This is because the Wasserstein distance that is commonly used in distributional data analysis poses challenges for multivariate distributions. A promising alternative is the sliced Wasserstein distance, which offers a computationally simpler solution. We propose distributional regression models with multivariate distributions as responses paired with Euclidean vector predictors, working with the sliced Wasserstein distance, which is based on a slicing transform from the multivariate distribution space to the sliced distribution space. We introduce two regression approaches, one based on utilizing the sliced Wasserstein distance directly in the multivariate distribution space, and a second approach that employs a univariate distribution regression for each slice. We develop both global and local Fr\'echet regression methods for these approaches and establish asymptotic convergence for sample-based estimators. The proposed regression methods are illustrated in simulations and by studying joint distributions of systolic and diastolic blood pressure as a function of age and joint distributions of excess winter death rates and winter temperature anomalies in European countries as a function of a country's base winter temperature.