  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.
