  Let S be a complex smooth projective surface and L be a line bundle on S.
G\"ottsche conjectured that for every integer r, the number of r-nodal curves
in |L| is a universal polynomial of four topological numbers when L is
sufficiently ample. We prove G\"ottsche's conjecture using the algebraic
cobordism group of line bundles on surfaces and degeneration of Hilbert schemes
of points. In addition, we prove the the G\"ottsche-Yau-Zaslow Formula which
expresses the generating function of the numbers of nodal curves in terms of
quasi-modular forms and two unknown series.
