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We complete the classification of compact Hermitian manifolds admitting aflat Gauduchon connection. In particular, we establish a conjecture of Yang andZheng, 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 holdswhen the flatness assumption is replaced by the so-called K\"ahler-likecondition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. Wealso treat the non-compact case.

Hermitian manifolds with flat Gauduchon connections