Vine copula models are a flexible tool in multivariate non-Gaussian distributions. For data from an observational study where the explanatory variables and response variables are measured over sampling units, we propose a vine copula regression method that uses regular vines and handles mixed continuous and discrete variables. This method can efficiently compute the conditional distribution of the response variable given the explanatory variables. Furthermore, we provide a theoretical analysis of the asymptotic conditional cumulative distribution function and quantile functions arising from vine copulas. Assuming all variables have been transformed to standard normal, the conditional quantile function could be asymptotically linear, sublinear, or constant with respect to the explanatory variables in different extreme directions, depending on the dependence properties of bivariate copulas in joint tails. The performance of the proposed method is evaluated on simulated data sets and a real data set. The experiments demonstrate that the vine copula regression method is superior to linear regression in making inferences with conditional heteroscedasticity.