The closed n-dimensional disc is the simplest smooth compact n-manifold and yet, despite continuous efforts of geometers and topologists since the beginning of the 60s, its group of symmetries (the topological group of diffeomorphisms) is still little understood. Over time it has become apparent that, although rooted in geometry and topology, the study of these groups is closely linked to several other areas of mathematics. In this talk, after giving a general introduction to the subject aimed at a broad audience, I will outline some of these connections and survey recent advances in the study of diffeomorphism groups of discs in relation to algebraic K-theory, exotic Pontryagin classes, and graph complexes à la Kontsevich.