This thesis examines the domination number of the semi-strong product of two graphs G and H where both G and H are simple and connected graphs. The product has an edge set that is the union of the edge set of the direct product of G and H together with the cardinality of V(H), copies of G. Unlike the other more common products (Cartesian, direct and strong), the semi-strong product is neither commutative nor associative. The semi-strong product is not supermultiplicative, so it does not satisfy a Vizing like conjecture. It is also not submultiplicative so it shares these two properties with the direct product. After giving the basic definitions related with graphs, domination in graphs and basic properties of the semi-strong product, this paper includes a general upper bound for the domination of the semi-strong product of any two graphs G and H as less than or equal to twice the domination numbers of each graph individually. Similar general results for the semi-strong product perfect-paired domination numbers of any two graphs G and H, as well as semi-strong products of some specific types of cycle graphs are also addressed.