In gauge theory, the prominent examples of enumerative invariants are Donaldson polynomials and Seiberg-Witten invariants, which help to distinguish different smooth structures on 4 dimensional manifolds. In recent years, other 4-manifold invariants have been introduced by changing the gauge theory (the PDE’s and the “counting” problem) or, by changing the dimension, similar gauge theory invariants were defined on higher-dimensional manifolds. Notable examples include Donaldson-Thomas (DT) invariants for six-dimensional, Calabi-Yau, manifolds. The study of enumerative geometry (counting of algebraic subspaces) of complex surfaces and threefolds has proven to be deeply related to physical structures, e.g. around Gopakumar-Vafa invariants (GV); Gromov-Witten invariants, DT, as well as Pandharipande-Thomas (PT) invariants; and their “motivic lifts". On the other hand, physical dualities in Gauge and String theory, such as Montonen-Olive duality and heterotic/Type II duality have also been a rich source of spectacular predictions about these counting invariants. An example of this is the modularity properties of GW or DT invariants which is proved mathematically in some cases, as suggested by the heterotic/Type II duality.