  Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index,
associated to a suitable ideal triangulation of a 3-manifold with torus
boundary components. The 3D-index is a collection of formal power series in
$q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is
a generating series of normal surfaces associated to the ideal triangulation.
This shows a connection of the 3D-index with classical normal surface theory,
and fulfills a dream of constructing topological invariants of 3-manifolds
using normal surfaces.
