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Many biologists are calling for an 'extended evolutionary synthesis' that would 'modernize the modern synthesis' of evolution. Biological information is typically considered as being transmitted across generations by the DNA sequence alone, but accumulating evidence indicates that both genetic and non-genetic inheritance, and the interactions between them, have important effects on evolutionary outcomes. We review the evidence for such effects of epigenetic, ecological and cultural inheritance and parental effects, and outline methods that quantify the relative contributions of genetic and non-genetic heritability to the transmission of phenotypic variation across generations. These issues have implications for diverse areas, from the question of missing heritability in human complex-trait genetics to the basis of major evolutionary transitions.