This volume contains the written account of the Bonn seminar on arith-metic geometry 2003/2004. It gives a coherent exposition of the theory of intersections of modular correspondences. The focus of the seminar is the formula for the intersection number of arithmetic modular correspondences due to Gross and Keating. Other topics treated are Hurwitz's theorem on the intersection of modular correspondences over the field of complex numbers, and the relation of the arithmetic intersection numbers to Fourier coefficients of Siegel-Eisenstein series. Also included is background material on one-dimensional formal groups and their endomorphisms, and on quadratic forms over the ring of p-adic integers.