Summary: Professor John Blake spent a considerable part of his scientific career on studying bubble dynamics and acoustic cavitation. As Blake was a mathematician, we will be focusing on the theoretical and numerical studies (and much less on experimental results). Rather than repeating what is essentially already known, we will try to present the results from a different perspective as much as possible. This review will also be of interest to readers who wish to know more about the boundary element method in general, which is a method often used by Blake and his colleagues to simulate bubbles. We will, however, not limit the discussion to bubble dynamics but try to give a broad discussion on recent advances and improvements to this method, especially for potential problems (Laplace) and wave equations (Helmholtz). Based on examples from Blake’s work, we will guide the reader and show some of the mysteries of bubble dynamics, such as why jets form in collapsing bubbles near rigid surfaces. Where appropriate, we will illustrate the concepts with examples drawn from numerical simulations and experiments.