  We extend the framework of K-stability (Tian, Donaldson) to more general
algebro-geometric setting, such as partial desingularisations of (fixed)
singularities, (not necessarily flat) families over higher dimensional base and
the classical birational geometry of surfaces.
  We also observe that "concavity" of the volume function implies decrease of
the (generalised) Donaldson-Futaki invariants along the Minimal Model Program,
in our generalised settings. Several related results on the connection with the
MMP theory, some of which are new even in the original setting of families over
curves, are also proved.
