In this article, we consider a non-parametric Bayesian approach to\nmultivariate quantile regression. The collection of related conditional\ndistributions of a response vector Y given a univariate covariate X is modeled\nusing a Dependent Dirichlet Process (DDP) prior. The DDP is used to introduce\ndependence across x. As the realizations from a Dirichlet process prior are\nalmost surely discrete, we need to convolve it with a kernel. To model the\nerror distribution as flexibly as possible, we use a countable mixture of\nmultidimensional normal distributions as our kernel. For posterior\ncomputations, we use a truncated stick-breaking representation of the DDP. This\napproximation enables us to deal with only a finitely number of parameters. We\nuse a Block Gibbs sampler for estimating the model parameters. We illustrate\nour method with simulation studies and real data applications. Finally, we\nprovide a theoretical justification for the proposed method through posterior\nconsistency. Our proposed procedure is new even when the response is\nunivariate.\n