This paper develops the Correlation Networks (CorNet) architecture for the extreme multi-label text classification (XMTC) task, where the objective is to tag an input text sequence with the most relevant subset of labels from an extremely large label set. XMTC can be found in many real-world applications, such as document tagging and product annotation. Recently, deep learning models have achieved outstanding performances in XMTC tasks. However, these deep XMTC models ignore the useful correlation information among different labels. CorNet addresses this limitation by adding an extra CorNet module at the prediction layer of a deep model, which is able to learn label correlations, enhance raw label predictions with correlation knowledge and output augmented label predictions. We show that CorNet can be easily integrated with deep XMTC models and generalize effectively across different datasets. We further demonstrate that CorNet can bring significant improvements over the existing deep XMTC models in terms of both performance and convergence rate. The models and datasets are available at: https://github.com/XunGuangxu/CorNet.