In this talk I will discuss the recent developments in the inverse spectral theory of bounded planner domains with strictly convex smooth boundaries. I will first present a joint work with Steve Zelditch in which we prove ellipses of small eccentricity are spectrally unique among all smooth domains. I will then discuss an inverse spectral result for nearly circular domains with an axial symmetry. A linearized version of this problem was studied by De Simoi, Kaloshin, and Wei. The non-linear problem is more interesting and uses second variations of the length functions and also some techniques of Avila, De Simoi, and Kaloshin