Quantile regression (QR) is a ubiquitous and powerful tool for estimating one or more conditional quantiles of a target variable Y given explanatory features X. A limitation of QR is that it is only defined for scalar target variables, due to the formulation of its objective function, and since the notion of quantiles has no standard definition for multivariate distributions. In this talk, I will review an optimal transport-based formulation of quantile estimation and regression and show a scalable extension to the multivariate case. I will show how the proposed framework is amenable to GPU-accelerated solvers for linear and nonlinear vector QR which maintain a fixed memory footprint with the number of samples and quantile levels, and demonstrate that they scale to millions of samples and thousands of quantile levels. I am particularly interested in novel applications of this tool.