This paper proposes a unified distributed robust regression framework for distributed massive data, which can include many robust regressions in one setting. Specifically, we first transfer different types of robust regressions into an asymptotically equivalent least-squares problem. Then the resulting estimator can be calculated as a weighted average of robust local estimators, and the communication cost is reduced, since it involves only one round of communication. In addition, since the local data information is incorporated sufficiently, it is adaptive to the heterogeneity. The new estimator is proven to be equivalent with the corresponding global robust regression estimator. Furthermore, we conduct variable selection based on the unified robust regression framework and adaptive LASSO, and the path of solution can also be conveniently obtained by LARS algorithm. It is theoretically shown that the new variable selection method can select true relevant variables consistently by using a new distributed BIC-type tuning parameter selector. The simulation results confirm the effectiveness of the new methods and the correctness of the theoretical results.