In this talk, we will look at several problems related to inscribed polygons in closed curves. The most famous of these is the inscribed square problem: Given a closed curve in the plane, do there exist four points on it forming the vertices of a square? But there are many other related problems that are also interesting. Of particular note is a theorem I proved together with a student, about inscribed triangles in closed curves. Given any triangle ABC and any closed curve in the plane, it is not too hard to show that there exist three points on the curve that form a triangle similar to ABC. But the corresponding question about curves in n-dimensional space is trickier. We proved that, under mild hypotheses, this is still true.