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In this thesis we will be looking at braid groups, their representations and how they play a role in quantum mechanics in two dimensions. We introduce configuration spaces, and prove a theorem by Artin relating the Artin braid groups $B_n$ to braids on $\R^2$. We consider the Burau representation and the Lawrence-Krammer-Bigelow representation and give an idea of their derivations. We discuss the relevance of braid groups for anyons in two dimensional space, and discuss their realization in the Fractional Quantum Hall Effect. Finally, we see how we might obtain braid group representations from category theory.