The Modave Summer School on Mathematical Physics is a yearly summer school in topics of theoretical physics ranging from quantum gravity and cosmology to theoretical particle physics and string theory. For this 2021 edition, the school exceptionally takes place in Brussels as opposed to its traditional venue in Modave, a charming village in the Belgian countryside close to Huy. The Modave School is organised by PhD students for PhD students, and this makes it rather unique. The courses are taught by Post-Docs or late PhD students, and they are all made of pedagogical, basic blackboard lectures about recent or fundamental topics in theoretical physics. Lectures of the seventeen edition are centered around the following subjects: Integrability in field theory, Horizon-scale microstructure of black holes, Goldstone boson physics and Interplays between cosmology, string theory and the Swampland program.