Abstract: I'll introduce and define eigenvalues of the Dirichlet and Neumann Laplacians. These eigenvalues are the frequencies of standing waves, and have an intriguing, elusive relationship with the geometry of the domain. In 1911, Weyl proved a result now known as "Weyl's law," showing that the asymptotic distribution of these frequencies encodes the volume of the domain. In 1961, Polya proved a much sharper result for domains that tile Euclidean space, bounding individual eigenvalues in terms of the volume of the domain, and conjectured that this bound holds for general domains. I'll state Weyl's law and sketch its proof, then spend most of the talk sketching a proof of Polya's theorem. Then I will state my recent result generalizing Polya's theorem and indicate its proof. Finally, I will discuss interesting open questions, including an analogue for Neumann eigenvalues and Polya's conjecture.