In this paper, the ψ-type stability and robustness of recurrent neural networks are investigated by using the differential inequality. By utilizing ψ-type functions combined with the inequality techniques, some sufficient conditions ensuring ψ-type stability and robustness are derived for linear neural networks with time-varying delays. Then, by choosing appropriate Lipschitz coefficient in subregion, some algebraic criteria of the multiple ψ-type stability and robust boundedness are established for the delayed neural networks with time-varying delays. For special cases, several criteria are also presented by selecting parameters with easy implementation. The derived results cover both ψ-type mono-stability and multiple ψ-type stability. In addition, these theoretical results contain exponential stability, polynomial stability, and μ-stability, and they also complement and extend some previous results. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed criteria.