  The $1/N$ expansion of matrix models is asymptotic, and it requires
non-perturbative corrections due to large $N$ instantons. Explicit expressions
for large $N$ instanton amplitudes are known in the case of Hermitian matrix
models with one cut, but not in the multi-cut case. We show that the recent
exact results on topological string instanton amplitudes provide the
non-perturbative contributions of large $N$ instantons in generic multi-cut,
Hermitian matrix models. We present a detailed test in the case of the cubic
matrix model by considering the asymptotics of its $1/N$ expansion, which we
obtain at relatively high genus for a generic two-cut background. These results
can be extended to certain non-conventional matrix models which admit a
topological string theory description. As an application, we determine the
large $N$ instanton corrections for the free energy of ABJM theory on the
three-sphere, which correspond to D-brane instanton corrections in superstring
theory. We also illustrate the applications of topological string instantons in
a more mathematical setting by considering orbifold Gromov-Witten invariants.
By focusing on the example of $\mathbb{C}^3/\mathbb{Z}_3$, we show that they
grow doubly-factorially with the genus and we obtain and test explicit
asymptotic formulae for them.
