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We embed polarised orbifolds with cyclic stabiliser groups into weightedprojective space via a weighted form of Kodaira embedding. Dividing by the(non-reductive) automorphisms of weighted projective space then formally givesa moduli space of orbifolds. We show how to express this as a reductivequotient and so a GIT problem, thus defining a notion of stability fororbifolds.  We then prove an orbifold version of Donaldson's theorem: the existence of anorbifold Kahler metric of constant scalar curvature implies K-semistability.  By extending the notion of slope stability to orbifolds we therefore get anexplicit obstruction to the existence of constant scalar curvature orbifoldKahler 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 ofGauntlett-Martelli-Sparks-Yau) fit into this framework.

Weighted projective embeddings, stability of orbifolds and constant scalar curvature K\"ahler metrics