We present CORA, a Cross-lingual Open-Retrieval Answer Generation model that can answer questions across many languages even when language-specific annotated data or knowledge sources are unavailable. We introduce a new dense passage retrieval algorithm that is trained to retrieve documents across languages for a question.Combined with a multilingual autoregressive generation model, CORA answers directly in the target language without any external translation or in-language retrieval modules as used in prior work. To train CORA, we extend available annotated data using an iterative training approach that improves retrieval for languages with limited resources. Our results show that CORA substantially outperforms the previous state-of-the-art model on multilingual open question answering benchmarks across 26 languages, 9 of which are unseen during training. Our analyses show the significance of cross-lingual retrieval and generation in many languages, particularly under low-resource settings. We will publicly release our code and model.