In these notes we provide a summary of recent results on the cohomological properties of
compact complex manifolds not endowed with a Kähler structure. On the one hand the large
number of developed analytic techniques makes it possible to prove strong cohomological
properties for compact Kähler manifolds. On the other in order to further investigate any of
these properties it is natural to look for manifolds that do not have any Kähler structure. We
focus in particular on studying Bott-Chern and Aeppli cohomologies of compact complex
manifolds. Several results concerning the computations of Dolbeault and Bott-Chern cohomologies
on nilmanifolds are summarized allowing readers to study explicit examples. Manifolds endowed
with almost-complex structures or with other special structures (such as for example
symplectic generalized-complex etc.) are also considered.
