Summary: A sequential labeling of a simple graph \(G\) (non tree) with \(m\) edges is an injective labeling \(f\) such that the vertex labels \(f(x)\) are from \(\{0,1,\dots ,m-1\}\) and the edge labels induced by \(f(x)+f(y)\) for each edge \(xy\) are distinct consecutive positive integers. A graph is sequential if it has a sequential labeling. We give some properties of sequential labeling and the criterion to verify sequential labeling. Necessary and sufficient conditions are obtained for every case of sequential graphs. A complete characterization of non-tree sequential graphs is obtained by vertex closure. Also, characterizations of sequential trees are given. The structure of sequential graphs is revealed.