The human auditory system extracts rich linguistic abstractions from speech signals. Traditional approaches to understanding this complex process have used linear feature-encoding models, with limited success. Artificial neural networks excel in speech recognition tasks and offer promising computational models of speech processing. We used speech representations in state-of-the-art deep neural network (DNN) models to investigate neural coding from the auditory nerve to the speech cortex. Representations in hierarchical layers of the DNN correlated well with the neural activity throughout the ascending auditory system. Unsupervised speech models performed at least as well as other purely supervised or fine-tuned models. Deeper DNN layers were better correlated with the neural activity in the higher-order auditory cortex, with computations aligned with phonemic and syllabic structures in speech. Accordingly, DNN models trained on either English or Mandarin predicted cortical responses in native speakers of each language. These results reveal convergence between DNN model representations and the biological auditory pathway, offering new approaches for modeling neural coding in the auditory cortex.