Explore adaptive wavelet methods and their applications in this comprehensive lecture by Rob Stevenson. Discover how to construct bases for Sobolev spaces on general domains using wavelets, and learn how these bases can be applied to optimally solve well-posed linear and nonlinear operator equations adaptively. Delve into various applications, including time-dependent PDEs, and gain insights into tensor product approximation and the reformulation of 2nd order PDEs as 1st order systems. This 86-minute presentation, part of the Hausdorff Trimester Program Multiscale Problems: Winter School on Numerical Analysis of Multiscale Problems, offers a deep dive into advanced mathematical concepts and their practical implementations.