This thesis investigates two possible versions of a "spectral curve" construction for compact constant mean curvature (CMC) surfaces of genus g > 1 in [Special characters omitted.] . The first version uses the holonomy spectral curve which was originally formulated for tori in [Special characters omitted.] . In order to make sense of the definition of this curve for a higher genus surface M , we must assume that the holonomy is abelian, and in this case it is shown that M must be a branched immersion factoring holomorphically through a CMC torus which can be located naturally in the Jacobian of M. The second version uses a curve defined as a double cover of M branched at the zeroes of the Hopf differential Q which coincides with that used originally by Hitchin to analyze the moduli space of stable bundles over M. We propose a method of defining a CMC immersion of this curve which has abelian holonomy and therefore, by the earlier result, factors through a naturally defined CMC torus. Along with the non-abelian holonomy of a certain meromorphic connection around the zeroes of Q , this data might provide effective moduli for M.