In this paper, multiple ψ-type stability of Cohen-Grossberg neural networks (CGNNs) with both time-varying discrete delays and distributed delays is investigated. By utilizing ψ-type functions combined with a new ψ-type integral inequality for treating distributed delay terms, some sufficient conditions are obtained to ensure that multiple equilibrium points are ψ-type stable for CGNNs with discrete and distributed delays, where the distributed delays include bounded and unbounded delays. These conditions of CGNNs with different output functions are less restrictive. More specifically, the algebraic criteria of the generalized model are applicable to several well-known neural network models by taking special parameters, and multiple different output functions are introduced to replace some of the same output functions, which improves the diversity of output results for the design of neural networks. In addition, the estimation of relative convergence rate of ψ-type stability is determined by the parameters of CGNNs and the selection of ψ-type functions. As a result, the existing results on multistability and monostability can be improved and extended. Finally, some numerical simulations are presented to illustrate the effectiveness of the obtained results.