Abstarct: A non-empty set of vertices is called a defensive alliance if has at least as many vertices from its closed neighbor in as it has in . A non-empty set of vertices is called a offensive alliance if each vertex in has more neighbors in than in . we consider alliances that are both defensive and offensive, which we call powerful alliances. Set is secure if and only if every attack on is defendable. We organize this thesis as follows: In chapter one we present elementary and basic concepts which we need in next chapters. In chapter two we study all kinds of alliances in graphs. In chapter three we study security in graphs. In chapter four we present our new results on open alliance in graphs, that listed as an open question by S.T. Hedetniem et al.