This thesis consists of two main parts. In the second part, we recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. W
e then show how to construct new equivariant analogues of these spectral sequences for RO(G)-graded Bredon homology and cohomology. Finally, we use these spectral sequences to complete a sample calculation, in which we use the equivariant Serre spectral sequence and the equivariant cohomology of complex projective spaces to compute the cohomology of the equivariant classifying space B_Cp O(2). However, to complete this sample computation, we need to know the cohomology of complex projective space. This calculation was done in a 1988 paper by Gaunce Lewis, but relies on a theorem whose proof as given was incorrect. We spend the first part of this thesis providing a correct proof and summarizing the results of Lewiss paper.
