Projective geometry provides a common framework for the study of classical Euclidean, spherical, and hyperbolic geometry. A major difference with the classical case is that a projective structure is not completely determined by its holonomy representation. In general, a complete description of the space of structures with the same holonomy is still missing. We will consider certain structures on punctured surfaces, and we will discuss how to describe all of those with a given holonomy in the case of the thrice-punctured sphere. This is done in terms of a certain geometric surgery known as grafting. Our approach involves a study of the Möbius completion, and of certain meromorphic differentials on Riemann surfaces. This is joint work with Sam Ballas, Phil Bowers, and Alex Casella.