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Identifying the processes by which human cultures spread across different populations is one of the most topical objectives shared amongst different fields of study. Seminal works have analysed a variety of data and attempted to determine whether empirically observed patterns are the result of demic and/or cultural diffusion. This special issue collects papers exploring several themes (from modes of cultural transmission to drivers of dispersal mechanisms) and contexts (from the Neolithic in Europe to the spread of computer programming languages), which offer new insights that will augment the theoretical and empirical basis for the study of demic and cultural diffusion. In this introduction we outline the state of art in the modelling of these processes, briefly discuss the pros and cons of two of the most commonly used frameworks (i.e. equation-based models and agent-based models), and summarise the significance of each paper published in this special issue.