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.