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The authors develop in detail the theory of (almost) c-projective geometry, a natural analogue of projective differential geometry adapted to (almost) complex manifolds. The authors realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kahler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kahler metrics underlying a given c-projective structure has many ramifications, which the authors explore in depth. As a consequence of this analysis, they prove the Yano-Obata Conjecture for complete Kahler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric.