Most existing scene text detectors require large-scale training data which cannot scale well due to two major factors: 1) scene text images often have domain-specific distributions; 2) collecting large-scale annotated scene text images is laborious. We study domain adaptive scene text detection, a largely neglected yet very meaningful task that aims for optimal transfer of labelled scene text images while handling unlabelled images in various new domains. Specifically, we design SCAST, a subcategory-aware self-training technique that mitigates the network overfitting and noisy pseudo labels in domain adaptive scene text detection effectively. SCAST consists of two novel designs. For labelled source data, it introduces pseudo subcategories for both foreground texts and background stuff which helps train more generalizable source models with multi-class detection objectives. For unlabelled target data, it mitigates the network overfitting by co-regularizing the binary and subcategory classifiers trained in the source domain. Extensive experiments show that SCAST achieves superior detection performance consistently across multiple public benchmarks, and it also generalizes well to other domain adaptive detection tasks such as vehicle detection.