Node polynomials for curves on surfaces

This text is a presentation of a set of formulae, first found by Vainsencher (for $\delta \le 6$) and shortly after improved by Kleiman and Piene, counting $\delta$-nodal curves in a complete linear system on a smooth surface, if $\delta \le 8$ and the corresponding line bundle is sufficiently positive. We also discuss a complement by Qviller, and related results due to Kazarian, Ohmoto, and others.