Abstract: Energy functions are used widely in the study of neural networks. It is known that if a neural network has a decreasing energy function bounded below and the equilibrium points of the network are identical with the zero points of the derivative of the energy function, then the network is stable and the equilibrium points are the local minimum points of the energy function. Examples are given in this paper to show that these results are actually not correct. It is proved under the above conditions for the energy functions that a neural network is stable if and only if all the solutions of the network are bounded. Relationships between the equilibrium points of neural networks and the local minimum points of energy functions are discussed. As applications, the stability of Hopfield neural network is proved rigorously and a neural network is given for finding out all eigenvectors of real symmetric matrix corresponding to the largest eigenvalue.