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We propose definitions of fairness in machine learning and artificial intelligence systems that are informed by the framework of intersectionality, a critical lens from the legal, social science, and humanities literature which analyzes how interlocking systems of power and oppression affect individuals along overlapping dimensions including gender, race, sexual orientation, class, and disability. We show that our criteria behave sensibly for any subset of the set of protected attributes, and we prove economic, privacy, and generalization guarantees. Our theoretical results show that our criteria meaningfully operationalize AI fairness in terms of real-world harms, making the measurements interpretable in a manner analogous to differential privacy. We provide a simple learning algorithm using deterministic gradient methods, which respects our intersectional fairness criteria. The measurement of fairness becomes statistically challenging in the minibatch setting due to data sparsity, which increases rapidly in the number of protected attributes and in the values per protected attribute. To address this, we further develop a practical learning algorithm using stochastic gradient methods which incorporates stochastic estimation of the intersectional fairness criteria on minibatches to scale up to big data. Case studies on census data, the COMPAS criminal recidivism dataset, the HHP hospitalization data, and a loan application dataset from HMDA demonstrate the utility of our methods.