  We complete the classification of compact Hermitian manifolds admitting a
flat Gauduchon connection. In particular, we establish a conjecture of Yang and
Zheng, showing that apart from the cases of a flat Chern or Bismut connection,
such manifolds are K\"ahler. More generally, we prove the same result holds
when the flatness assumption is replaced by the so-called K\"ahler-like
condition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. We
also treat the non-compact case.
