Two very closely related definitions of robustness of a sequence of estimators are given which take into account the types of deviations from parametric models that occur in practice. These definitions utilize the properties of the Prokhorov distance between probability distributions. It is proved that weak $^\ast$-continuous functionals on the space of probability distributions define robust sequences of estimators (in either sense). The concept of the "breakdown point" of a sequence of estimators is defined, and some examples are given.