Abstract: Dilute Bose gases are unique quantum systems that exhibit a fascinating low-temperature phase known as the Bose-Einstein condensate. Over the past two decades, the mathematical understanding of these systems has improved considerably. In this talk, we will review some of these advances, with a perspective on the largely open challenge of understanding their general behaviour in the thermodynamic limit, the appropriate large-scale framework for investigating the occurrence of phase transitions. Based on a series of joint works with G. Basti, C. Boccato, C. Brennecke, A. Giuliani, A. Olgiati, G. Pasqualetti and B. Schlein. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page