The equivariant cohomology of a manifold M acted upon by a compact Lie group G is defined to be the singular cohomology groups of the topological space (M × EG)/G. It is well known that the equivariant cohomology of M is parametrised by the Cartan model of equivariant differential forms. However, this model has no obvious geometric interpretation – partly because the expression above is not a manifold in general. Work in the 70s by Segal, Bott and Dupont indicated that this space can be constructed as the geometric realisation of a simplicial manifold that is naturally built out of M and G. This simplicial manifold carries a complex of so-called simplicial differential forms which gives a much more natural geometric interpretation of differential forms on the topological space (M × EG)/G. This thesis provides a model for the equivariant cohomology of a manifold in terms of this complex of simplicial differential forms. Explicit chain maps are constructed, inducing isomorphisms on cohomology, between this complex of simplicial differential forms and the more standard models of equivariant cohomology, namely the Cartan and Weil models. ; Thesis (M.Phil.) -- University of Adelaide, School of Mathematical Sciences, 2017.