abstract: In this talk we will discuss some properties of the moduli spaces of (possibly) branched \( \mathbb{CP}^1 \)-structures on surfaces. Examples of such structures are hyperbolic metric with cone-singularities multiple of \(2\pi\) and translation surfaces. We will focus in local/global surgeries that allow to travel in the moduli space of such structures. (Joint workwith G. Calsamiglia and B. Deroin).