  We explain how structures analogous to those appearing in the theory of
stability conditions on abelian and triangulated categories arise in geometric
invariant theory. This leads to an axiomatic notion of a central charge on a
scheme with a group action, and ultimately to a notion of a stability condition
on a stack analogous to that on an abelian category. We use these ideas to
introduce an axiomatic notion of a stability condition for polarised schemes,
defined in such a way that K-stability is a special case.
  In the setting of axiomatic geometric invariant theory on a smooth projective
variety, we produce an analytic counterpart to stability and explain the role
of the Kempf-Ness theorem. This clarifies many of the structures involved in
the study of deformed Hermitian Yang-Mills connections, Z-critical connections
and Z-critical K\"ahler metrics.
