In 1911 Toeplitz conjectured that any simple continuous closed curve in the plane inscribes a square. A less famous variant of this problem is Hadwiger's 1971 conjecture that any simple closed continuous curve in 3-space inscribes a parallelogram. Both conjectures have been resolved under some smoothness condition on the curve. I will survey some of the known results and then report on recent progress on both conjectures. This is joint work with Jai Aslam, Shujian Chen, Sam Saloff-Coste, Linus Setiabrata, and Hugh Thomas.