A complex projective structure on a surface S is a geometric structure modelled on the Riemann sphere. The monodromy of the structure determines a representation from the fundamental group of S to PSL(2,C), which is the group of conformal automorphisms of the Riemann sphere. The study of the moduli space of such structures, and what monodromy representations arise, has had a long history, arising from the theory of linear differential equations on the complex plane, and has connections with topology, complex analysis and hyperbolic geometry. In this talk I will describe some of these connections, and survey known results, including some recent work of mine with various collaborators, about the case when S is a punctured surface.