We propose a notion of conditional vector quantile function and a vector quantile regression. A conditional vector quantile function (CVQF) of a random vector, taking values in given covariates , taking values in , is a map , which is monotone, in the sense of being a gradient of a convex function, and such that given that vector follows a reference non-atomic distribution , for instance uniform distribution on a unit cube in , the random vector has the distribution of conditional on . Moreover, we have a strong representation, almost surely, for some version of . The vector quantile regression (VQR) is a linear model for CVQF of given . Under correct specification, the notion produces strong representation, , for denoting a known set of transformations of , where is a monotone map, the gradient of a convex function and the quantile regression coefficients have the interpretations analogous to that of the standard scalar quantile regression. As becomes a richer class of transformations of , the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge–Kantorovich’s optimal transportation problem at its core as a special case. In the classical case, where is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.