In recent years, there has been substantial progress in the mathematical understanding of the macroscopic behaviour of dilute Bose gases. In particular, the validity of a celebrated formula for the second-order asymptotic of the ground state energy of dilute bosons — first predicted by Lee, Huang, and Yang in 1957 — has been fully established in the case of integrable (non-negative) interactions. In the first part of my talk, I will present the main ideas behind the recent results, the relevant length scales of the problem, and the open questions that lie ahead. I will then discuss how a simple trial state, introduced by Bijl-Dingle-Jastrow in the 1950s, can be used to derive an upper bound for the ground state energy of a dilute Bose gas of hard sphere, which captures the Lee-Huang-Yang expansion up to the order of the sub-leading correction. An upper bound that establishes the Lee-Huang-Yang formula for hard spheres is, in fact, still missing. Based on a series of works with G. Basti, A. Giuliani, A. Olgiati, G. Pasqualetti and B. Schlein.