  The essence of the second law of classical thermodynamics is the `entropy
principle' which asserts the existence of an additive and extensive entropy
function, S, that is defined for all equilibrium states of thermodynamic
systems and whose increase characterizes the possible state changes under
adiabatic conditions. It is one of the few really fundamental physical laws (in
the sense that no deviation, however tiny, is permitted) and its consequences
are far reaching. This principle is independent of models, statistical
mechanical or otherwise, and can be understood without recourse to Carnot
cycles, ideal gases and other assumptions about such things as `heat',
`temperature', `reversible processes', etc., as is usually done. Also the well
known formula of statistical mechanics, S = -\sum p log p, is not needed for
the derivation of the entropy principle.
  This contribution is partly a summary of our joint work (Physics Reports,
Vol. 310, 1--96 (1999)) where the existence and uniqueness of S is proved to be
a consequence of certain basic properties of the relation of adiabatic
accessibility among equilibrium states. We also present some open problems and
suggest directions for further study.
