The problem considered is the robust estimation of the variance parameter in a heteroscedastic linear model. We treat the situation in which the variance is a function of the explanatory variables. To estimate robustly the variance in this case, it is necessary to guard against the influence of outliers in the design as well as outliers in the response. By analogy with the homoscedastic regression case, we propose two estimators that do this. Their performances are evaluated on a number of data sets. We had considerable success with estimators that bound the “self-influence”—that is. the influence an observation has on its own fitted value. We conjecture that in other situations (e.g., homoscedastic regression) bounding the selfinfluence will lead to estimators with good robustness properties.