  We prove that every irreducible component of the coarse
Koll\'ar-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log
Calabi-Yau surfaces is a finite quotient of a projective toric variety. This
verifies a conjecture of Hacking-Keel-Yu. The proof combines tools from log
smooth deformation theory, the minimal model program, punctured log
Gromov-Witten theory and mirror symmetry.
