In this talk, we recall definitions of some (well-known) metrics on a compact complex manifold X, such as K\"ahler, balanced, SKT and LCK and H-s. Moreover, the relations between these metrics will be investigated. We show that the K\ahler metrics are a special case of all the above metrics, but the converse is not true. Finally, we recall the notion of \partial\bar\patial-manifolds and prove that the Bott-Chern and the Aeppli cohomology groups are isomorphic under this assumption.