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Language is the best example of a cultural evolutionary system, able to retain a phylogenetic signal over many thousands of years. The temporal stability (conservatism) of basic vocabulary is relatively well understood, but the stability of the structural properties of language (phonology, morphology, syntax) is still unclear. Here we report an extensive Bayesian phylogenetic investigation of the structural stability of numerous features across many language families and we introduce a novel method for analyzing the relationships between the "stability profiles" of language families. We found that there is a strong universal component across language families, suggesting the existence of universal linguistic, cognitive and genetic constraints. Against this background, however, each language family has a distinct stability profile, and these profiles cluster by geographic area and likely deep genealogical relationships. These stability profiles seem to show, for example, the ancient historical relationships between the Siberian and American language families, presumed to be separated by at least 12,000 years, and possible connections between the Eurasian families. We also found preliminary support for the punctuated evolution of structural features of language across families, types of features and geographic areas. Thus, such higher-level properties of language seen as an evolutionary system might allow the investigation of ancient connections between languages and shed light on the peopling of the world.