A p-Kähler structure on a complex manifold is a closed, transverse (p,p)-form. For the extremal values of p, we get the well known Kähler and balanced manifolds, however, for each other possible p, such structures have no metric meaning. The aim of the talk, after discussing similarities and differences between p-Kähler structures and Kähler and balanced structures, is to present some results on the existence of p-Kähler structures, with a focus on Lie groups and their compact quotients. Based on a work in progress with A. Fino and G. Grantcharov.