This thesis investigates the structure of the projective coordinate rings of SL(n,C) weight varieties. An SL(n,C) weight variety is a Geometric Invariant Theory quotient of the space of full flags by the maximal torus in SL(n,C). Special cases include configurations of n-tuples of points in projective space modulo automorphisms of projective space. There are three main results. The first is an explicit finite set of generators for the coordinate ring. The second is that the lowest degree elements of the coordinate ring provide a well-defined map from the weight variety to projective space. The third theorem is an explicit presentation for the ring of projective invariants of n ordered points on the Riemann sphere, in the case that each point is weighted by an even integer. The methods applied involve matroid theory and degenerations of the weight varieties to toric varieties attached to Gelfand Tsetlin polytopes.