Abstract: It is known that an important property of the discrete Hopfield\|type neural network is that it always converges to a stable state when operating in a serial mode and to a cycle of length at most 2 when operating in a full parallel model.These properties are the basis for the potential applications of this model,such as associative memory devices and combinatorial optimization.Convergence theorems of discrete Hopfield\|type neural networks with delay are obtained in the paper.Under a proper assumption,it is proved that any discrete Hopfield\|type neural network with delay will converge to a stable state when operating in the serial mode,and one of the weight matrices is a symmetric one and can generalize convergence theorem in earlier works.The authors also relate maximum of modified energy function to stable state of neural network with delay and obtain evolution features in neighborhood of stable state.In other words,this network can converge to a stable state after one time interval.Accordant relations between convergence of the energy function and stabilization of correspondent network in the serial mode are presented as well.