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.
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