A polyhedral graph G is called PCC if every vertex of G has strictly positive combinatorial curvature and the graph is not a prism or antiprism. In this thesis it is shown that the maximum order of a 3-regular PCC graph is 132 and the 3-regular PCC graphs which match that bound are enumerated. A new PCC graph with two 39-faces and 208 vertices is constructed, matching the number of vertices of the largest PCC graphs discovered by Nicholson and Sneddon. A conjecture that there are no PCC graphs with faces of size larger than 39 is made, along with a proof that if there are no faces of size larger than 122, then there is an upper bound of 244 on the order of PCC graphs. / Graduate