This paper addresses the robust stability of recurrent neural networks (RNNs) with time-varying delays and input perturbation, where the time-varying delays include discrete and distributed delays. By employing the new ψ-type integral inequality, several sufficient conditions are derived for the robust stability of RNNs with discrete and distributed delays. Meanwhile, the robust boundedness of neural networks is explored by the bounded input perturbation and L¹-norm constraint. Moreover, RNNs have a strong anti-jamming ability to input perturbation, and the robustness of RNNs is suitable for associative memory. Specifically, when input perturbation belongs to the specified and well-characterized space, the results cover both monostability and multistability as special cases. It is revealed that there is a relationship between the stability of neural networks and input perturbation. Compared with the existing results, these conditions proposed in this paper improve and extend the existing stability in some literature. Finally, the numerical examples are given to substantiate the effectiveness of the theoretical results.