This 71st volume of the Bourbaki Seminar gathers the texts of the fifteen survey lectures delivered during the year 2018/2019. Among the topics addressed the reader will find: Euclidean lattices and theta series, pluripotential theory, random regular graphs, non-exact groups and spaces, the companion conjecture, $C^\ast$-simplicity, birational geometry of Fano varieties, stable reduction in higher dimension, higher rank Teichmüller theories, HOMFLY polynomials and Hilbert schemes, Riemann zeta function and probability, phase transition in percolation, triangulation conjecture, homology of Hurwitz spaces, minimal hypersurfaces.