Graph theory is of great importance in various elds of science. Simple nite graphs represent a signicant type of graph. This nal paper discusses the degree sequence of a nite simple graph, a concept closely related to the basic elements of a graph, i.e., vertices and edges. Two well-known criteria which give the necessary and sucient conditions for a sequence of nonnegative numbers to be a degree sequence of a graph have been proved, the Havel-Hakimi theorem and the Erd®s-Gallai theorem. The Havel-Hakimi theorem is based on the residual sequence of the largest element of a degree sequence and the Erd®s-Gallai theorem gives the most known explicit characterization of a degree sequence. Based on the Havel-Hakimi theorem, an algorithm can be developed to determine whether a set of nonnegative numbers is a degree sequence. The concept of a degree set of a graph was considered and a theorem was proved to determine the smallest number of vertices of a graph among all graphs of which it is a degree set. A degree sequence of a signed graph was considered and the criteria were specied to make the degree sequence a graphical sequence.