This is a survey of the theory of complex projective ℂℙ1-structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics:
- Using the Schwarzian derivative to parameterize the moduli space
- Thurston’s parameterization of the moduli space using grafting
- Holonomy representations of ℂℙ1-structures
We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map.