Hilbert's eleventh problem asks about sums of squares (or more generally representability by quadratic forms) of integers in a number field. Recent progress has led to its solution. In the first lecture we will give an introductory lecture to this problem and the key steps that led to its solution. In the second lecture we will describe (also at an introductory level) the spectral theory of modular surfaces indicating where this is used to solve the problem in lecture 1. We also explain how this spectral theory is of interest in quantum chaos.