Can One Hear the Shape of a Drum? Revisited

In a landmark paper, Mark Kac in 1966 [Amer. Math. Monthly, 73, pp. 1–23] showed that geometric properties of regions in R² can be obtained by studying the asymptotic properties of the spectrum of the Laplacian. He also conjectured that the shape of a region might be completely determined by the spectrum. We describe recent developments in this field and discuss the status of the Kac conjecture as well as similar conjectures for operators other than the Laplacian.