Polya's spectral conjecture for the Dirichlet Laplacian states that its eigenvalues are always above the first term in the corresponding Weyl asymptotics. We will begin by providing a fairly comprehensive introduction to the problem, which dates back to the first edition of Pólya's 1954 book Mathematics and plausible reasoning: patterns of plausible inference, and then present some recent results. In particular, we will discuss relations to extremal eigenvalue problems, extensions to manifolds, and some specific examples illustrating the different possible behaviour related to both the low- and high-frequency regimes.
