We will discuss the Gross-Zagier theorem for modular curves and introduce its high dimensional generalizations, both over number fields and over function fields. The main objects of the talks are special values of L-functions and their connection to arithmetic intersection numbers of modular cycles, i.e., special algebraic cycles on the moduli space of abelian varieties with certain additional structures. In the end we discuss the arithmetic fundamental lemma conjecture which connects intersection numbers of modular cycles over p-adic fields to lattice-counting problems.