I briefly describe five old but still actively explored problems of the Spectral Theory of Partial Differential Equations
1. How eigenvalues are distributed (where eigenvalues often mean squares of the frequencies in the mechanical or electromagnetic problems or energy levels in the quantum mechanics models) and the relation to the behaviour of the billiard trajectories.
2. Equidistribution of eigenfunctions and connection to ergodicity of billiard trajectories (a quantum chaos and a classical chaos).
3. Can one hear the shape of the drum?
4. Nodal lines and Chladni plates.
5. Strange spectra of quantum systems.
Dmitry Jakobson made several very useful remarks and suggestions