This article aims at analyzing and designing the multivalued high-capacity-associative memories based on recurrent neural networks with both asynchronous and distributed delays. In order to increase storage capacities, multivalued activation functions are introduced into associative memories. The stored patterns are retrieved by external input vectors instead of initial conditions, which can guarantee accurate associative memories by avoiding spurious equilibrium points. Some sufficient conditions are proposed to ensure the existence, uniqueness, and global exponential stability of the equilibrium point of neural networks with mixed delays. For neural networks with ${n}$ neurons, ${m}$ -dimensional input vectors, and ${2k}$ -valued activation functions, the autoassociative memories have ${(2k)^{n}}$ storage capacities and heteroassociative memories have min ${\{(2k)^{n},(2k)^{m}\}}$ storage capacities. That is, the storage capacities of designed associative memories in this article are obviously higher than the ${2^{n}}$ and min ${\{2^{n},2^{m}\}}$ storage capacities of the conventional ones. Three examples are given to support the theoretical results.