This article addresses complexity by selecting some of its key aspects that share a common feature: the power to change. They seem to change not only the way the world is approached by scientists, but also the way this approach, the resulting perspectives and their multiple relationships, are interpreted. These main aspects are: (1) the challenge of measurability, with an unexpected result that escapes the gravitational field of the measurability problem; (2) the meaning of reproducibility and the redrawn boundaries of scientific inquiry, with implications for the social sciences; (3) the altered expectations concerning prediction, which seem to break with a glorious tradition of unquestioned technological success; and (4) the discovery of all-embracing patterns of events that unavoidably include large events, possibly perceived as ‘crises’, which one may hope to understand and confront, rather than rule out. The resulting geography, with its new landmarks, new relationships among its elements and new means of orientation, is expected to reach the public sooner or later, even if the effect – according to complexity theory itself – cannot be foreseen in detail. All these fibres of change are considered in the context of a fresh meaning of time and of a topology dominated by network concepts.