Both exponential and stochastic stabilities of the Hopfield neural network are analyzed. The results are especially useful for analyzing the stabilities of asymmetric neural networks. A constraint on the connection matrix has been found under which the neural network has a unique and exponentially stable equilibrium. Given any connection matrix, this constraint can be satisfied through the adjustment of the gains of the amplifiers and the resistances in the neural net circuit. A one-to-one and smooth map between input currents and the equilibria of the neural network can be set up. The above results can be applied to the master/slave net to prove that the master net can find the best connection matrix for the slave net. For the neural network disturbed by some noise, the stochastic stability of the network is also analyzed. A special asymmetric neural network formed by a closed chain of formal neurons is also studied for its stability and oscillation. Both stable and oscillatory dynamics are obtained in the closed chain network through the adjustment of the gains and resistances of the amplifiers.