Human languages evolve by a process of cultural evolution in which linguistic varieties compete for our attention and adapt to the environment of our minds. This has produced a remarkable cultural replicator whose fidelity of transmission can rival that of some genes. Yet progress in understanding this defining human trait is impeded by an influential view that the processes of linguistic change are so rapid, idiosyncratic and many that general evolutionary rules and laws describing language evolution and language histories will not be found. That perspective can be contrasted with results emerging from recent empirical studies that point to simple and general rules of language evolution that have held throughout human history. This research programme is designed to bridge the gap in these perspectives. Our goal is to build and apply to real-world data, statistical models of linguistic evolution that will put the study of language change on a footing comparable to that of molecular evolution.Micro-evolutionary studies will formulate and test population-linguistic models of cultural selection acting on words competing within populations of speakers. Macro-evolutionary or comparative studies will build a novel probabilistic model of sound changes and apply it to reconstructing ancestral lexical forms (the proto-language) and language histories. Shared-process studies link these two levels, investigating how common patterns of word-usage and sound changes mediate language evolution around the globe. Our studies are relevant to questions of language competition and extinction and we will investigate how words evolve, reconstruct proto-languages and infer deep linguistic histories -- such as the linguistic settlement of Eurasia, and search for evidence of a Mother Tongue.