In this talk, my goal is to explain what it means to degenerate the complex structure of a Riemann surface and how this fits in the study of the moduli space of complex flat connections. I will start by recalling what a Riemann surface is and explain why their study has been central in geometry until now. Then, I will give an intuition of what degenerating their complex structure means, and I hope to introduce the audience to the question that drives me in my current research.