The number of nodal curves on algebraic surfaces has been studied for over 100 years. On the projective space P^2, it asks how many homogenous degree d polynomials passing through some general points has only nodes as singularities. On K3, the numbers was predicted by the famous Yau-Zaslow formula using physics argument.

It is conjectured that for sufficiently ample line bundles, the numbers of nodal curves in |L| is given by universal polynomials of L^2, LK, and the first and second Chern numbers of the surface. By universal we mean the polynomial is independent of the line bundle and surface.

In this talk, I will present a proof of this universality theorem. This result gives another proof of the Yau-Zaslow formula. Furthermore, we prove Goettsche's conjectural generating function using quasi-modular forms for the numbers.