Distortion is a geometric knot invariant introduced by Gromov in the 1980s. It measures the smallest bi-Lipschitz constant required to embed the knot in Euclidean space. In this talk, I'll discuss the question: given a knot, how do we estimate its distortion? This question has been very difficult even though distortion has an elementary definition. I'll explain why, and I'll explain a new approach to systematically understand this question. For fibered knots, this approach translates questions about knot distortion into combinatorial questions about curves on surfaces.