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 method is complemented with flexible types of penalties (Lasso, elastic-net, group lasso, sparse group lasso, scad and mcp) to deal with complex low-dimensional structures.