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Multilingual question answering tasks typically assume answers exist in the same language as the question. Yet in practice, many languages face both information scarcity -- where languages have few reference articles -- and information asymmetry -- where questions reference concepts from other cultures. This work extends open-retrieval question answering to a cross-lingual setting enabling questions from one language to be answered via answer content from another language. We construct a large-scale dataset built on questions from TyDi QA lacking same-language answers. Our task formulation, called Cross-lingual Open Retrieval Question Answering (XOR QA), includes 40k information-seeking questions from across 7 diverse non-English languages. Based on this dataset, we introduce three new tasks that involve cross-lingual document retrieval using multi-lingual and English resources. We establish baselines with state-of-the-art machine translation systems and cross-lingual pretrained models. Experimental results suggest that XOR QA is a challenging task that will facilitate the development of novel techniques for multilingual question answering. Our data and code are available at https://nlp.cs.washington.edu/xorqa.