Explore the dimension theory of self-affine systems in this 44-minute lecture from the Simons Semester on Dynamics series. Delve into the complexities of calculating Hausdorff dimensions for planar self-affine sets, focusing on cases involving strong separation of cylinders and strong irreducibility of linear parts. Learn how the well-understood principles of conformal transformations in iterated function systems (IFS) become more challenging when dealing with non-conformal maps and affine transformations. Examine fundamental results from Falconer and recent developments based on collaborative research with Antti Kaenmaki, Mike Hochman, and Ariel Rapaport.