Abstract: Associated to a family of curves $C\to S$ are $\ell$-adic monodromy representations, which generalize Galois representations. I will discuss part my ongoing thesis work demonstrating a big monodromy result for the moduli space of superelliptic curves. This result uses an arithmeticity result of reduced Burau representations of Venkataramana and clutching methods of Achter and Pries. Time permitting, I will also describe applications of this big monodromy result in other parts of my thesis --- it can be used to prove a Cohen-Lenstra result for function fields and to prove a result on the vanishing of zeta functions for Kummer curves over the projective line over finite fields.