Transformer-based models have spurred unprecedented increases in performance on NLP tasks. As these models' performance begins to approach (or even surpass) that of humans, the question of how they achieve this performance becomes crucial. Do these models acquire the same linguistic knowledge, and learn to perform the same type of linguistic processing underlying human language? Or, is their high performance driven more by other factors, unrelated to the mechanisms driving human language processing? This Ph.D. aims to leverage techniques and domain knowledge from cognitive science and linguistics in order to understand both the behavior and the internal mechanisms at the heart of these models. Doing so has the potential to not only increase the robustness and generalization of such models but also shed light on language acquisition and processing in humans as well.