In manufacturing systems there has been major advances made in the design, scheduling, operation and performance. However, in resource sharing systems deadlock still occurs. All deadlock avoidance methods developed only provide sufficient conditions for deadlock. These methods cannot specify which parts to move in the system to avoid deadlock. In this dissertation the concepts of order, slack, space, and evaluation state are employed to precisely quantify both necessary and sufficient conditions for deadlock. It is shown that a manufacturing system that is in an evaluation state is deadlock-free if and only if a set of inequalities is satisfied. An algorithm approach is used to prove these results. This algorithm can be employed to specify which parts to move to avoid deadlock. Several examples showing the theory are presented and are compared with other methods.