  This survey covers recent developments on the geometry and physics of
Looijenga pairs, namely pairs $(X,D)$ with $X$ a complex algebraic surface and
$D$ a singular anticanonical divisor in it. I will describe a surprising web of
correspondences linking together several a priori distant classes of
enumerative invariants associated to $(X,D)$, including the log Gromov--Witten
invariants of the pair, the Gromov--Witten invariants of an associated higher
dimensional Calabi--Yau variety, the open Gromov--Witten invariants of certain
special Lagrangians in toric Calabi--Yau threefolds, the Donaldson--Thomas
theory of a class of symmetric quivers, and certain open and closed BPS-type
invariants. I will also discuss how these correspondences can be effectively
used to provide a complete closed-form solution to the calculation of all these
invariants.
