  In this paper:
  1. We give a complete classification of Hermitian non-K\"ahler ALF/AF
gravitational instantons. This confirms a conjecture by Aksteiner-Andersson.
  2. We also show that such gravitational instantons can be compactified to
toric log del Pezzo surfaces. This explains a curious relation between
Hermitian non-K\"ahler ALF/AF gravitational instantons and compact Hermitian
non-K\"ahler Einstein 4-manifolds.
  3. We prove the non-existence of ALG and ALH type Hermitian non-K\"ahler
gravitational instantons.
