This paper develops the asymptotic theory for least absolute deviation estimation of a shift in linear regressions. Rates of convergence and asymptotic distributions for the estimated regression parameters and the estimated shift point are derived. The asymptotic theory is developed both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. Asymptotic distributions are also obtained for trending regressors and for dependent disturbances. The analysis is carried out in the framework of partial structural change, allowing some parameters not to be influenced by the shift. Efficiency relative to least-squares estimation is also discussed. Monte Carlo analysis is performed to assess how informative the asymptotic distributions are.