In this paper, the novel robust maximum entropy clustering algorithm RMEC, as the improved version of the maximum entropy algorithm MEC, is presented to overcome its drawbacks: very sensitive to outliers and uneasy to label them. With the introduction of Vapnik's ε-insensitive loss function and the new weight factors, the new objective function is re-constructed, and consequently, its new update rules are derived according to the Lagrangian optimization theory. Compared with algorithm MEC, the main contributions of algorithm RMEC exist in its much better robustness for outliers and the fact that it can effectively label outliers in the dataset using the obtained weight factors. The experimental results demonstrate its superior performance in enhancing the robustness and labeling outliers in the dataset.