Curve Counting \`a la G\"ottsche

Let n_\delta be the number of \delta-nodal curves lying in a suitably ample complete linear system |L| and passing through appropriately many points on a smooth projective complex algebraic surface. A major open problem is to understand the behavior of n_\delta, specifically to finish off Lothar G"ottsche’s mostly proved 1997 conjectures and then go on to treat the new refinements by G"ottsche and Vivek Shende. Five subproblems are explained, and work on them is surveyed.