We employ distribution regression to estimate the joint distribution of two outcome variables conditional on covariates. Bivariate Distribution Regression (BDR) is particularly valuable when some dependence between the outcomes persists after accounting for the impact of the covariates. Our analysis relies on Chernozhukov et al. (2018) which shows that any conditional joint distribution has a local Gaussian representation. We describe how BDR can be implemented and present some functionals of interest. As modeling the unexplained dependence is a key feature of BDR, we focus on functionals related to this dependence. We decompose the difference between the joint distributions for different groups into composition, marginal and sorting effects. We provide a similar decomposition for the transition matrices which describe how location in the distribution of one outcome is associated with location in the other. Our theoretical contributions are the derivation of the properties of these estimated functionals and appropriate procedures for inference. Our empirical illustration focuses on intergenerational mobility. Using the Panel Survey of Income Dynamics data, we model the joint distribution of parents’ and children’s earnings.