  Microcanonical Thermodynamics allows the application of Statistical Mechanics
on one hand to closed finite and even small systems and on the other to the
largest,self-gravitating ones. However, one has to reconsider the fundamental
principles of Statistical Mechanics especially its key quantity, entropy.
Whereas in conventional Thermostatistics the homogeneity and extensivity of the
system and the concavity of its entropy S(E) are central conditions, these fail
for the systems considered here. E.g. at phase separation the entropy S(E) is
necessarily convex to make e^{S(E)-E/T} bimodal in E (the two coexisting
phases). This is so even for normal macroscopic systems with short-range
coupling. As inhomogeneities and surface effects in particular cannot be scaled
away,one has to be careful with the standard arguments of splitting a system
into two or bringing two systems into thermal contact. Not only the volume part
of the entropy must be considered. When removing an external constraint in
regions of a negative heat capacity, the system may even relax under a flow of
heat (energy) against the temperature slope. Thus Clausius formulation of the
Second Law: "Heat always flows from hot to cold" can be violated. Temperature
is not a necessary or fundamental control parameter of Thermostatistics. In the
final sections of this paper the general microscopic mechanism leading to
condensation and to the convexity of the microcanonical entropy S(E) at phase
separation is sketched. Also the microscopic conditions for the existence or
non-existence of a critical end-point of the phase-separation are discussed.
This is explained for the liquid--gas and the solid--liquid transition.
