The extensions of Boltzmann-Gibbs classical entropic form are presented. The focus is on the thermodynamical entropies which do not obey classical definition of additivity and extensity with main motivation coming from the gravitational systems admitting long-range interactions both in astrophysics (black holes) and in cosmology. Horizon entropies are the main concern. In the paper, the plethora of non-Boltzmann-Gibbs entropies are studied and the comparative analysis of their properties with respect to additivity and extensivity is presented.