Given an abelian scheme over char 0 and an irreducible subvariety X, one can define the t-th degeneracy locus of X for each integer t. This geometric concept of degeneracy loci has recently seen many applications in Diophantine Geometry, notably when t is 0 and 1, in the recent developments on the uniformity of the number of rational points on curves, on the solutions of the Uniform Mordell-Lang Conjecture and of the Relative Manin-Mumford Conjecture. In this talk, I will define the degeneracy loci in the universal abelian variety, and explain how they are used in the applications mentioned above.