The periodic oscillation and the global exponential stability of a class of recurrent neural networks with non-monotone activation functions and time-varying delays are analyzed. For two sets of activation functions, some algebraic criteria for ascertaining global exponential periodicity and global exponential stability of the class of recurrent neural networks are derived by using the comparison principle and the theory of monotone operator. These conditions are easy to check in terms of system parameters. In addition, we provide a new and efficacious method for the qualitative analysis of various neural networks. The periodic oscillation and the global exponential stability of a class of recurrent neural networks with non-monotone activation functions and time-varying delays are analyzed. For two sets of activation functions, some algebraic criteria for ascertaining global exponential periodicity and global exponential stability of the class of recurrent neural networks are derived by using the comparison principle and the theory of monotone operator.