There are many interesting and widely used estimators of a functional with ﬁnite semi-parametric variance bound that depend on nonparametric estimators of nuisance func-tions. We use cross-ﬁtting to construct such estimators with fast remainder rates. We give cross-ﬁt doubly robust estimators that use separate subsamples to estimate diﬀerent nuisance functions. We show that a cross-ﬁt doubly robust spline regression estimator of the expected conditional covariance is semiparametric eﬃcient under minimal conditions. Corresponding estimators of other average linear functionals of a conditional expectation are shown to have the fastest known remainder rates under certain smoothness conditions. The cross-ﬁt plug-in estimator shares some of these properties but has a remainder term that is larger than the cross-ﬁt doubly robust estimator. As speciﬁc examples we consider the expected conditional covariance, mean with randomly missing data, and a weighted average derivative.