  We discuss how quantitative cohomological informations could provide
qualitative properties on complex and symplectic manifolds. In particular we
focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since
they represent useful tools in studying non K\"ahler geometry. We give an
overview on the comparisons among the dimensions of the cohomology groups that
can be defined and we show how we reach the $\partial\overline\partial$-lemma
in complex geometry and the Hard-Lefschetz condition in symplectic geometry.
For more details we refer to [6] and [29].
