Abstract: The stability of neural networks is known to be basis of the network’s various applications In this paper, the dynamic behavior of discrete Hopfield neural networks with delay is mainly studied by using the state transition equation and the energy function The sufficient conditions, under which the networks are convergent to cycle of length of 1 or 2, are present The sufficient conditions for the networks being convergent to special cycle of length of 2 and 4 are given Also, the conditions for the networks having no any stable states are achieved The obtained results not only generalize some existing results, but also provide a theoretical foundation of performance analysis and new applications of the networks