  In this thesis we study the relationship between the existence of canonical
metrics on a complex manifold and stability in the sense of geometric invariant
theory. We introduce a modification of K-stability of a polarised variety which
we conjecture to be equivalent to the existence of an extremal metric in the
polarisation class. A variant for a complete extremal metric on the complement
of a smooth divisor is also given. On toric surfaces we prove a Jordan-Holder
type theorem for decomposing semistable surfaces into stable pieces. On a ruled
surface we compute the infimum of the Calabi functional for the unstable
polarisations, exhibiting a decomposition analogous to the Harder-Narasimhan
filtration of an unstable vector bundle.
