%0 Journal Article %T Complex Projective Structures %A David Dumas %J Mathematics %D 2009 %I arXiv %X This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the moduli space (2) Thurston's parameterization of the moduli space using grafting (3) Holonomy representations of CP^1 structures We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map. %U http://arxiv.org/abs/0902.1951v1