Jointly organized with the RTG Berlin-Hannover "From geometry to numbers: Moduli, Hodge theory, rational points" financed by the DFG. Based on the 4 lecture series of 4 lectures each, the Simons School will highlight the enumerative and birational geometry of the moduli spaces of curves, abelian varieties and K3 surfaces, presenting modern techniques of geometric, combinatorial and tropical nature that have led to decisive progress. Part of the thematic semester Algebraic Geometry (August-December 2026) hosted by the Erdős Center.