Neural network pruning can be effectively applied to compress automatic speech recognition (ASR) models. However, in multilingual ASR, performing language-agnostic pruning may lead to severe performance degradation 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 (-5.0% average WER) and a language-agnostically pruned model (-21.4% average WER), and provide better performance on low-resource languages compared to the monolingual sparse models.