The Katz-Klemm-Vafa conjecture expresses the GromovWitten
theory of K3 surfaces (and K3-fibred 3-folds in fibre classes)
in terms of modular forms. Its recent proof gives the first non-toric
geometry in dimension greater than 1 where Gromov-Witten theory is
exactly solved in all genera.
We survey the various steps in the proof. The MNOP correspondence
and a new Pairs/Noether-Lefschetz correspondence for K3-fibred
3-folds transform the Gromov-Witten problem into a calculation of the
full stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.