It is a well-known fact that nilmanifolds do not admit any Kähler metric, with the exception of complex tori. Therefore, if one wants to generalize the Kähler condition (e.g. balanced, SKT...), they appear as a natural class where to look for this sort of "weaker" structures. In this talk, we will review some general facts about nilmanifolds and recall some special types of Hermitian metrics, paying special attention to the recently introduced class of k-th Gauduchon ones.