Robustness is a desirable property for many statistical techniques. As an important measure of robustness, breakdown point has been widely used for regression problems and many other settings. Despite the existing development, we observe that the standard breakdown point criterion is not directly applicable for many classification problems. In this paper, we propose a new breakdown point criterion, namely angular breakdown point, to better quantify the robustness of different classification methods. Using this new breakdown point criterion, we study the robustness of binary large margin classification techniques, although the idea is applicable to general classification methods. Both bounded and unbounded loss functions with linear and kernel learning are considered. These studies provide useful insights on the robustness of different classification methods. Numerical results further confirm our theoretical findings.