Data are often contaminated by outliers. Classical robust regression methods introduce robustness by sacrificing statistical accuracy and efficiency, which is less desirable. In this talk, we discuss our recent work on adaptive robust regression methods, which simultaneously achieve robustness, asymptotic unbiasedness and full asymptotic efficiency. The key idea is that the robustification parameter should diverge to infinity in a data-dependent manner. We focus on two adaptive robust regression methods which are the adaptive versions of the classical Huber regression and the redescending regression. We examine the performances of these methods both theoretically and numerically. We also discuss other possible applications and extensions.