In this talk, I am going to review recent progress in the mathematical understanding of the low-energy properties of dilute quantum systems. In particular, I am going to present a rigorous version of Bogoliubov theory and I am going to show how it can be used to obtain precise estimates for the ground state energy and the low-energy excitation spectrum and also to approximate the time evolution of Bose gases in the so-called Gross-Pitaevskii regime.