In the last 30 years there has been growing interest in and a greater appreciation of the unique contributions that medieval authors have made to the history of logic. In this thesis, we compare and contrast the modal logics of Robert Kilwardby and John Buridan and explore how their two conceptions of modality relate to and differ from modern notions of modal logic. We develop formal reconstructions of both authors' logics, making use of a number of different formal techniques. In the case of Robert Kilwardby we show that using his distinction between per se and per accidens modalities, he is able to provide a consistent interpretation of the apodictic fragment of Aristotle's modal syllogism and that, by generalising this distinction to hypothetical construction, he can develop an account of connexive logic. In the case of John Buridan we show that his modal logic is a natural extension of the usual Kripke-style possible worlds semantics, and that this modal logic can be shown to be sound and complete relative to a proof-theoretic formalisation of Buridan's treatment of the expository syllogism.