"A Brief History of Geometry” is a overview of the main historical steps of Geometry, from Euclid to Perelman, via Descartes, Gauss, Riemann, Ricci, Poincaré and Einstein… This conference was given by Jean-Pierre Bourguignon (IHES, France) at the Academy of Sciences of Lisbon the November 2nd, 2017, at the invitation of the Centro de Matemática, Aplicações Fundamentais e Investigação Operacional da Faculdade de Ciências da Universidade de Lisboa, Portugal.