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The development of models of the relationship between biodiversity and ecosystem function (BEF) has advanced rapidly over the last 20 years, incorporating insights gained through extensive experimental work. We propose Generalised Diversity-Interactions models that include many of the features of existing models and have several novel features. Generalised Diversity-Interactions models characterise the contribution of two species to ecosystem function as being proportional to the product of their relative abundances raised to the power of a coefficient ?. A value of ? <1 corresponds to a stronger than expected contribution of species' pairs to ecosystem functioning, particularly at low relative abundance of species. Varying the value of ? has profound consequences for community-level properties of BEF relationships, including: (i) saturation properties of the BEF relationship; (ii) the stability of ecosystem function across communities; (iii) the likelihood of transgressive overyielding. For low values of ?, loss of species can have a much greater impact on ecosystem functioning than loss of community evenness. Generalised Diversity-Interactions models serve to unify the modelling of BEF relationships as they include several other current models as special cases. Generalised Diversity-Interactions models were applied to seven data sets and three functions: total biomass (five grassland experiments), community respiration (one bacterial experiment) and nitrate leaching (one earthworm experiment). They described all the nonrandom structure in the data in six experiments, and most of it in the seventh experiment and so fit as well or better than competing BEF models for these data. They were significantly better than Diversity-Interactions models in five experiments. Synthesis. We show that Generalized Diversity-Interactions models quantitatively integrate several methods that separately address effects of species richness, evenness and composition on ecosystem function. They describe empirical data at least as well as alternative models and improve the ability to quantitatively test among several theoretical and practical hypotheses about the effects of biodiversity levels on ecosystem function. They improve our understanding of important aspects of the relationship between biodiversity (evenness and richness) and ecosystem function (BEF), which include saturation, effects of species loss, the stability of ecosystem function and the incidence of transgressive overyielding.