  We embed polarised orbifolds with cyclic stabiliser groups into weighted
projective space via a weighted form of Kodaira embedding. Dividing by the
(non-reductive) automorphisms of weighted projective space then formally gives
a moduli space of orbifolds. We show how to express this as a reductive
quotient and so a GIT problem, thus defining a notion of stability for
orbifolds.
  We then prove an orbifold version of Donaldson's theorem: the existence of an
orbifold Kahler metric of constant scalar curvature implies K-semistability.
  By extending the notion of slope stability to orbifolds we therefore get an
explicit obstruction to the existence of constant scalar curvature orbifold
Kahler metrics. We describe the manifold applications of this orbifold result,
and show how many previously known results (Troyanov, Ghigi-Kollar,
Rollin-Singer, the AdS/CFT Sasaki-Einstein obstructions of
Gauntlett-Martelli-Sparks-Yau) fit into this framework.
