Abstract: Vector quantile regression (VQR) estimates the conditional vector quantile function (CVQF), a fundamental quantity which fully represents the conditional distribution of $\mathbf{Y}|\mathbf{X}$. VQR is formulated as an optimal transport (OT) problem between a uniform $\mathbf{U}\sim\mu$ and the target $(\mathbf{X},\mathbf{Y})\sim\nu$, the solution of which is a unique transport map, co-monotonic with $\mathbf{U}$. Recently NL-VQR has been proposed to estimate support non-linear CVQFs, together with fast solvers which enabled the use of this tool in practical applications. Despite its utility, the scalability and estimation quality of NL-VQR is limited due to a discretization of the OT problem onto a grid of quantile levels. We propose a novel _continuous_ formulation and parametrization of VQR using partial input-convex neural networks (PICNNs). Our approach allows for accurate, scalable, differentiable and invertible estimation of non-linear CVQFs.We further demonstrate, theoretically and experimentally, how continuous CVQFs can be used for general statistical inference tasks: estimation of likelihoods, CDFs, confidence sets, coverage, sampling, and more.This work is an important step towards unlocking the full potential of VQR.