  The Katz-Klemm-Vafa conjecture expresses the Gromov-Witten 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.

Algebraic Geometry;High Energy Physics - Theory