  Fujita and Li have given a characterisation of K-stability of a Fano variety
in terms of quantities associated to valuations, which has been essential to
all recent progress in the area. We introduce a notion of valuative stability
for arbitrary polarised varieties, and show that it is equivalent to
K-stability with respect to test configurations with integral central fibre.
The numerical invariant governing valuative stability is modelled on Fujita's
beta invariant, but includes a term involving the derivative of the volume. We
give several examples of valuatively stable and unstable varieties, including
the toric case. We also discuss the role that the delta invariant plays in the
study of valuative stability and K-stability of polarised varieties.
