  The aim of this focus letter is to present a comprehensive classification of
the main entropic forms introduced in the last fifty years in the framework of
statistical physics and information theory. Most of them can be grouped into
three families, characterized by two-deformation parameters, introduced
respectively by Sharma, Taneja, and Mittal (entropies of degree
$(\alpha,\,\beta$)), by Sharma and Mittal (entropies of order
$(\alpha,\,\beta)$), and by Hanel and Thurner (entropies of class $(c,\,d)$).
Many entropic forms examined will be characterized systematically by means of
important concepts such as their axiomatic foundations {\em \`{a} la}
Shannon-Khinchin and the consequent composability rule for statistically
independent systems. Other critical aspects related to the Lesche stability of
information measures and their consistency with the Shore-Johnson axioms will
be briefly discussed on a general ground.
