Punctuated equilibrium is a characteristic of some natural and social systems that occasionally generate bursts of activity at different scales following periods of stagnation of different durations. This phenomenon can be formalized as pink noise, which is an attribute of self-organized criticality (SOC). This paper outlines the main ideas of the SOC theory and the tools for identifying pink noise. It presents examples of punctuated equilibrium found in computer experiments with artificial societies, as well as in historical and political studies. Such examples are found, in particular, in demographic processes, in the development of markets, in the dynamics of electoral choice, in the Internet activity of network communities, in terrorist and criminal activity, and in protest movements in the past and present. The SOC theory explains why avalanche-like social transformations do not always occur under the influence of some major extraordinary factor. Social cataclysms can be caused by the intrinsic properties of systems as well as by micro-level processes and local impulses. Internal transformational potential can be high enough and, at the same time, subtle. The SOC theory describes the ways in which such avalanche conditions can be identified with mathematic rigor and, at the same time, relatively easily.