The moving sofa problem is a well-known open problem in geometry. Posed by Leo Moser in 1966, it asks for the planar shape of largest area that can be moved around a right-angled corner in a two-dimensional hallway of width 1. Although deceptively easy to state, it turns out to be highly nontrivial to analyze, and has a rich structure that is intriguing to amateurs and experts alike. In this talk I will survey both old and new results about the problem, including a new moving sofa shape with an interesting algebraic structure that I discovered in 2016, and new bounds on the area of a moving sofa I derived more recently in joint work with Yoav Kallus. I will conclude with a discussion of how the heavily experimental and computer-assisted nature of the recent results offers broader lessons for aspiring research mathematicians.