This paper presents new criteria for the existence of stable equilibrium points in the total saturation region for cellular neural networks (CNNs). It is shown that the results obtained can be used to derive some complete stability conditions for some special classes of  CNNs such as positive cell-linking CNNs, opposite-sign CNNs and dominant-template CNNs. Our results are  also compared with the previous results derived in the literature for the existence of stable equilibrium points for CNNs.