Abstract: Associative memory and optimization are two most important neural network functions. Asymptotically stable equilibria are the only network operating points which can be used to associative memory. In this paper, the asymptotic stability for nonlinear continuous associative memory neural networks is studied. Several important theorems guaranteeing the network's asymptotic stability are derived. These theorems are more general than the existing conclusions to ensure the network's asymptotic stability. On the basis of these theorems, it is discussed how to used these theorems to optimally, design associative memory neural networks. The effectiveness of the optimal design method proposed in this paper is theoretically proved.