  The principal methods for the definition of thermodynamic entropy are
discussed with special reference to those developed by Carath\'eodory, the
Keenan School, Lieb and Yngvason, and the present authors. An improvement of
the latter method is then presented. Seven basic axioms are employed: three
Postulates, which are considered as having a quite general validity, and four
Assumptions, which identify the domains of validity of the definitions of
energy (Assumption 1) and entropy (Assumptions 2, 3, 4). The domain of validity
of the present definition of entropy is not restricted to stable equilibrium
states. For collections of simple systems, it coincides with that of the proof
of existence and uniqueness of an entropy function which characterizes the
relation of adiabatic accessibility proposed by Lieb and Yngvason. However, our
treatment does not require the formation of scaled copies so that it applies
not only to collections of simple systems, but also to systems contained in
electric or magnetic fields and to small and few-particle systems.
