I investigate the asymptotic distribution of linear quantile regression coefficient estimates when the parameter may lie on the boundary of the parameter space, and related inference procedures when the null hypothesis asserts that the parameters lie on a boundary of this set. Particular attention is paid to parameter spaces defined by sets of linear inequalities. I provide a uniform characterization of the constrained quantile regression process over an interval of quantile levels. This asymptotic theory is used to derive asymptotic inference methods for three related processes based on the constrained quantile regression process.