There is a famous recursion formula by Kontsevich to find the number of 
genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points. 
My question is the following: Let $S$ be a complex surface and $A$ a fixed 
homology class in $H^2(S,\mathbb{Z})$. Is there any hope of answering this question:

``How many genus zero curves are there (through the right number of points) that 
represent the homology class $A$ in $S$?''

Does Kontsevich's argument rely heavily on the fact that $S$ is $\mathbb{P}^2$? 
Are there some large classes of $S$ for which his argument might go through? 

Note that to find ``the right number of points'' you have to calculate the 
dimension of the moduli space (for which there is a formula).