This paper provides a short review of recent results on synchronization in small-world dynamical networks of coupled oscillators. We also propose a new model of small-world networks of cells with a time-varying coupling and study its synchronization properties. It is shown that such a time-varying structure of the network can ensure more reliable synchronization than the conventional small-worlds. The term "small world" refers to a network of locally connected nodes having a few additional long-range shortcuts chosen at random. The addition of the shortcuts sharply reduces the average distance between the nodes and therefore provides the so-called small-world effect. Discovered first in social networks, the small-world effect appeared to be a characteristic of many real-world structure both human-generated or of biological origin. For social networks, this property implies that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. However, the structure of social networks is not homogeneous, there are always key persons that provide distant out-local-world connections between people. This paper is written in honor of the 60th birthday of our friend and colleague, Wadim S. Anischenko, who is one of such key persons in European Nonlinear Dynamics community.