The graph edge coloring problem is to color the edges of a graph such that adjacent edges receives different colors. Let $G$ be a simple graph with maximum degree $\Delta$. The minimum number of colors needed for such a coloring of $G$ is called the chromatic index of $G$, written $\chi'(G)$. We say $G$ is of class one if $\chi'(G)=\Delta$, otherwise it is of class 2. A majority of edge coloring papers is devoted to the Classification Problem for simple graphs. A graph $G$ is said to be \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. Hilton in 1985 conjectured that every graph $G$ of class two with $\Delta(G)>\frac{|V(G)|}{3}$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. In this thesis, I will introduce some of my researches toward the Classification Problem of simple graphs, and a new approach to the overfull conjecture together with some new techniques and ideas.