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tes mellitus and hypertension in Toronto.The Price equation describes the change in populations. Change concerns some value, such as biological fitness, information or physical work. The Price equation reveals universal aspects for the nature of change, independently of the meaning ascribed to values. By understanding those universal aspects, we can see more clearly why fundamental mathematical results in different disciplines often share a common form. We can also interpret more clearly the meaning of key results within each discipline. For example, the mathematics of natural selection in biology has a form closely related to information theory and physical entropy. Does that mean that natural selection is about information or entropy? Or do natural selection, information and entropy arise as interpretations of a common underlying abstraction? The Price equation suggests the latter. The Price equation achieves its abstract generality by partitioning change into two terms. The first term naturally associates with the direct forces that cause change. The second term naturally associates with the changing frame of reference. In the Price equation's canonical form, total change remains zero because the conservation of total probability requires that all probabilities invariantly sum to one. Much of the shared common form for the mathematics of different disciplines may arise from that seemingly trivial invariance of total probability, which leads to the partitioning of total change into equal and opposite components of the direct forces and the changing frame of reference. This article is part of the theme issue 'Fifty years of the Price equation'.The Price equation has been entangled with social evolution theory from the start. It has been used to derive the most general versions of kin selection theory, and Price himself produced a multilevel equation that provides an alternative formulation of social evolution theory, dividing selection into components between and within groups. In this sense, the Price equation forms a basis for both kin and group selection, so often pitted against each other in the literature. Contextual analysis and the neighbour approach are prominent alternatives for analysing group selection. I discuss these four approaches to social evolution theory and their connections to the Price equation, focusing on their similarities and common mathematical structure. Despite different notations and modelling traditions, all four approaches are ultimately linked by a common set of mathematical components, revealing their underlying unity in a transparent way. The Price equation can similarly be used in the derivation of streamlined, weak selection social evolution modelling methods. These weak selection models are practical and powerful methods for constructing models in evolutionary and behavioural ecology; they can clarify the causal structure of models, and can be easily converted between the four social evolution approaches just like their regression counterparts. This article is part of the theme issue 'Fifty years of the Price equation'.The Price equation was a piece of abstract mathematics. What kind of a connection could it possibly have had to George Price's personal life and biography? Here, I will argue that the initial impetus for Price's foray into mathematical population genetics stemmed from a preoccupation with the origins of family, one that was born following a divorce from his wife and the abandonment of their two young girls. What is special about the Price equation is the way in which it associates statistically between two groups, a 'mother' and 'daughter' population. The association need not mean genetic relatedness in the narrow sense of direct descent, and it allows us to see selection working at different levels simultaneously, a fact that was not lost on William Hamilton. Hamilton was one of the few friends who desperately tried to save Price from falling into the abyss of depression and homelessness in the period following the publication of 'Selection and covariance' (Price 1928 Nature 227, 520-521 (doi10.1038/227520a0)). Viewed in this light, the Price equation assumes new meaning. This article is part of the theme issue 'Fifty years of the Price equation'.The genetic response to selection is central to both evolutionary biology and animal and plant breeding. While Price's theorem (PT) is well-known in evolutionary biology, most breeders are unaware of it. Rather than using PT, breeders express response to selection as the product of the intensity of selection (i), the accuracy of selection (ρ) and the additive genetic standard deviation (σA); R = iρσA. In contrast to the univariate 'breeder's equation', this expression holds for multivariate selection on Gaussian traits. Here, I relate R = iρσA to PT, and present a generalized version, R = iwρA,wσA, valid irrespective of the trait distribution. Next, I consider genotype-environment covariance in relation to the breeder's equation and PT, showing that the breeder's equation may remain valid depending on whether the genotype-environment covariance works across generations. Finally, I consider the response to selection in the prevalence of an endemic infectious disease, as an example of an emergent trait. The result shows that disease prevalence has much greater heritable variation than currently believed. The example also illustrates that the indirect genetic effect approach moves elements of response to selection from the second to the first term of PT, so that changes acting via the social environment come within the reach of quantitative genetics. This article is part of the theme issue 'Fifty years of the Price equation'.Price's equation provides a very simple-and very general-encapsulation of evolutionary change. find more It forms the mathematical foundations of several topics in evolutionary biology, and has also been applied outwith evolutionary biology to a wide range of other scientific disciplines. However, the equation's combination of simplicity and generality has led to a number of misapprehensions as to what it is saying and how it is supposed to be used. Here, I give a simple account of what Price's equation is, how it is derived, what it is saying and why this is useful. In particular, I suggest that Price's equation is useful not primarily as a predictor of evolutionary change but because it provides a general theory of selection. As an illustration, I discuss some of the insights Price's equation has brought to the study of social evolution. This article is part of the theme issue 'Fifty years of the Price equation'.In this paper, I will argue that the generality of the Price equation comes at a cost, and that is that the terms in it become meaningless. There are simple linear models that can be written in a Price equation-like form, and for those the terms in them have a meaningful interpretation. There are also models for which that is not the case, and in general, when no assumptions on the shape of the fitness function are made, and all possible models are allowed for, the regression coefficients in the Price equation do not allow for a meaningful interpretation. The failure to recognize that the Price equation, although general, only has a meaningful interpretation under restrictive assumptions, has done real damage to the field of social evolution, as will be illustrated by looking at an application of the Price equation to group selection. This article is part of the theme issue 'Fifty years of the Price equation'.The diversity of genetic and non-genetic processes that make offspring resemble their parents are increasingly well understood. In addition to genetic inheritance, parent-offspring similarity is affected by epigenetic, behavioural and cultural mechanisms that collectively can be referred to as non-genetic inheritance. Given the generality of the Price equation as a description of evolutionary change, is it not surprising that the Price equation has been adopted to model the evolutionary implications of non-genetic inheritance. In this paper, we briefly introduce the heredity perspectives on which those models rely, discuss the extent to which these perspectives make different assumptions and place different emphases on the roles of heredity and development in evolution, and the types of empirical research programmes they motivate. The existence of multiple perspectives and explanatory aims highlight, on the one hand, the versatility of the Price equation and, on the other hand, the importance of understanding how heredity and development can be conceptualized in evolutionary studies. This article is part of the theme issue 'Fifty years of the Price equation'.The Price equation is widely recognized as capturing conceptually important properties of natural selection, and is often used to derive versions of Fisher's fundamental theorem of natural selection, the secondary theorems of natural selection and other significant results. However, class structure is not usually incorporated into these arguments. From the starting point of Fisher's original connection between fitness and reproductive value, a principled way of incorporating reproductive value and structured populations into the Price equation is explained, with its implications for precise meanings of (two distinct kinds of) reproductive value and of fitness. Once the Price equation applies to structured populations, then the other equations follow. The fundamental theorem itself has a special place among these equations, not only because it always incorporated class structure (and its method is followed for general class structures), but also because that is the result that justifies the important idea that these equations identify the effect of natural selection. The precise definitions of reproductive value and fitness have striking and unexpected features. However, a theoretical challenge emerges from the articulation of Fisher's structure is it possible to retain the ecological properties of fitness as well as its evolutionary out-of-equilibrium properties? This article is part of the theme issue 'Fifty years of the Price equation'.The Price equation shows that evolutionary change can be written in terms of two fundamental variables the fitness of parents (or ancestors) and the phenotypes of their offspring (descendants). Its power lies in the fact that it requires no simplifying assumptions other than a closed population, but realizing the full potential of Price's result requires that we flesh out the mathematical representation of both fitness and offspring phenotype. Specifically, both need to be treated as stochastic variables that are themselves functions of parental phenotype. Here, I show how new mathematical tools allow us to do this without introducing any simplifying assumptions. Combining this representation of fitness and phenotype with the stochastic Price equation reveals fundamental rules underlying multivariate evolution and the evolution of inheritance. Finally, I show how the change in the entire phenotype distribution of a population, not simply the mean phenotype, can be written as a single compact equation from which the Price equation and related results can be derived as special cases. This article is part of the theme issue 'Fifty years of the Price equation'.