The study of self-duality has attracted some attention over the past decade. A good deal of research in that time has been done on constructing and classifying all self-dual graphs and in particular polyhedra. We will give an overview of the recent research in the first two chapters. In the third chapter, we will show the necessary condition that a self-complementary self-dual graph have <i>n</i> ≡ 0, 1 (mod 8) vertices and we will review White's infinite class (the Paley graphs, for which <i>n</i> ≡ 1 (mod 8)). Finally, we will construct a new infinite class of self-complementary self-dual graphs for which <i>n</i> ≡ 0 (mod 8).