Abstract: A self-similar set is a compact set in Euclidean space which decomposes into similar scaled copies of itself, often disjoint from each other. For example – the middle-thirds Cantor set. The dimension theory of these sets has been (for the most part) understood for a long time. If one assumes instead that the scaled copies are related to the whole by an affine map rather than a similarities, the analysis is much harder, and results have mostly been for random (rather than individual) examples. Over the past three years the situation has changed radically and, under some very reasonable assumptions, is now completely solved in the plane. I will explain the main ingredients, which are a Ledrappier-Young type formula for the associated measures (due to Barany and Kaenmaki), and techniques from additive combinatorics developed in a joint paper with Barany and Rapaport.