Human cultural traits-behaviors, ideas, and technologies that can be learned from other individuals-can exhibit complex patterns of transmission and evolution, and researchers have developed theoretical models, both verbal and mathematical, to facilitate our understanding of these patterns. Many of the first quantitative models of cultural evolution were modified from existing concepts in theoretical population genetics because cultural evolution has many parallels with, as well as clear differences from, genetic evolution. Furthermore, cultural and genetic evolution can interact with one another and influence both transmission and selection. This interaction requires theoretical treatments of gene-culture coevolution and dual inheritance, in addition to purely cultural evolution. In addition, cultural evolutionary theory is a natural component of studies in demography, human ecology, and many other disciplines. Here, we review the core concepts in cultural evolutionary theory as they pertain to the extension of biology through culture, focusing on cultural evolutionary applications in population genetics, ecology, and demography. For each of these disciplines, we review the theoretical literature and highlight relevant empirical studies. We also discuss the societal implications of the study of cultural evolution and of the interactions of humans with one another and with their environment. cultural evolution | mathematical models | gene-culture coevolution | niche construction | demography H uman culture encompasses ideas, behaviors, and artifacts that can be learned and transmitted between individuals and can change over time (1). This process of transmission and change is reminiscent of Darwin's principle of descent with modification through natural selection, and Darwin himself drew this explicit link in the case of languages: "The formation of different languages and of distinct species, and the proofs that both have been developed through a gradual process, are curiously parallel" (2, 3). Theory underpins most scientific endeavors, and, in the 1970s, researchers began to lay the groundwork for cultural evolutionary theory, building on the neo-Darwinian synthesis of genetics and evolution by using verbal, diagrammatic, and mathematical models (4-8). These models are, by necessity, approximations of reality (9), but because they require researchers to specify their assumptions and extract the most important features from complex processes, they have proven exceedingly useful in advancing the study of cultural evolution (10). Here, we review the field of cultural evolutionary theory as it pertains to the extension of biology through culture. We focus on human culture because the bulk of cultural evolutionary models are human-centric and certain processes such as cumulative culture seem to be unique to humans. However, numerous nonhuman species also exhibit cultural transmission, and we consider the areas of overlap between models of human and animal culture in Discussion.The study of cultural evolution is im...