Estimation and inference for conditional linear quantile regression models using a convolution smoothed approach. In the low-dimensional setting, efficient gradient-based methods are employed for fitting both a single model and a regression process over a quantile range. Normal-based and (multiplier) bootstrap confidence intervals for all slope coefficients are constructed. In high dimensions, the conquer methods complemented with l_1-penalization and iteratively reweighted l_1-penalization are used to fit sparse models.