Humans understand speech with such speed and accuracy, it belies the complexity of transforming sound into meaning. The goal of my research is to develop a theoretically grounded, empirically tested and computationally explicit account of how the brain achieves this feat. In the work presented here, I overview a set of magneto-encephalography studies that describe (i) what linguistic representations the brain uses to bridge between sound and meaning; (ii) how those representations are combined to form hierarchical structures (e.g. phonemes into morphemes; morphemes into words); (iii) how information is exchanged across structures to guide comprehension from the bottom-up and top-down. The research also contributes to a broader analytical framework — informed by machine-learning and classic statistics — which allows neural signals to be decomposed into an interpretable sequence of operations. Overall, this dissertation showcases the utility of combining theoretical linguistics, machine-learning and cognitive neuroscience for developing empirically- and performance-optimised models of spoken language processing.