For a finite group G, the Hurwitz space Hir,ng (G) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G. Let εr(G) = {(x1,...,xr) : G = ⟨x1,...,xr⟩, Πri=1xi = 1, xi ϵ G#, i = 1,...,r}. The connected components of Hir,ng(G) are in bijection with braid orbits on εr(G).
In this thesis we enumerate the connected components of Hir,ng(G) in the cases where g≤ 2 and G is a primitive affine group. Our approach uses a combination of theoretical and computational tools. To handle the most computationally challenging cases we develop a new algorithm which we call the Projection-Fiber algorithm