Petri nets are used as models in the study of networks involving information flows. Petri nets have also turned out to be useful in the study of many asynchronous concurrent systems.In this thesis, the notions of deadlocks, traps, and liveness are considered from a graph theoretic viewpointA characterization of minimal deadlocks and traps in Petri nets is obtained. For the complete Petri nets, alternative characterizations of deadlocks and traps are obtained. Necessary and sufficient conditions are obtained for complete Petri nets to be deadlock free. Similar conditions for trap free complete Petri nets are also determined.