As everyone knows from real-life experience, moving sofas around corners is often tricky: not every sofa shape will fit. The moving sofa problem is a mathematical question that elegantly captures the subtleties involved even in a simplified two-dimensional setting. It asks for the planar shape of maximal area that can be moved around a right-angled corner in a corridor of width 1. The problem has been open for 50 years, and has a complicated conjectured solution known as Gerver's sofa, proposed in 1992 - a shape whose boundary has 18 distinct pieces. In this talk I will explain the mathematics of this fascinating problem,

and tell about a new approach to the study of the problem that I developed recently, and some additional related results. The talk will be self-contained, will require no prerequisite knowledge beyond standard calculus, and will include many entertaining animations of moving sofas.