Small world networks have a peculiar property: even though almost all of their nodes are only locally connected, the average path length between two nodes is nearly as low as that of random networks. Networks of this type are ubiquitous in natural systems and frequently show interesting dynamics. We investigated the behavior of a new class of cellular nonlinear networks (CNNs) where all cells are locally connected to their nearest neighbors (as in a conventional CNN), but some of the nodes have long-range shortcuts as well. We give lower bounds on the number of extra links that must be added for a network to be considered as small-world and suggest an optimal topology for the added links. We also show some interesting behaviors of this network in diffusion and wave propagation phenomena.