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Neural network pruning compresses automatic speech recognition (ASR) models effectively. However, in multilingual ASR, language-agnostic pruning may lead to severe performance drops on some languages because language-agnostic pruning masks may not fit all languages and discard important language-specific parameters. In this work, we present ASR pathways, a sparse multilingual ASR model that activates language-specific sub-networks ("pathways"), such that the parameters for each language are learned explicitly. With the overlapping sub-networks, the shared parameters can also enable knowledge transfer for lower-resource languages via joint multilingual training. We propose a novel algorithm to learn ASR pathways, and evaluate the proposed method on 4 languages with a streaming RNN-T model. Our proposed ASR pathways outperform both dense models and a language-agnostically pruned model, and provide better performance on low-resource languages compared to the monolingual sparse models.