The Box–Cox transformation is a parametric power transformation technique introduced by Box and Cox (1964) to address issues such as non-additivity, non-normality, and heteroscedasticity in data. This review summarizes the development, applications, and variations of the Box–Cox transformation, as well as its use in statistical modeling and inference. The transformation is defined as a function of the data that depends on a parameter λ, which is estimated to best fit the data. For λ ≠ 0, the transformation is y^λ, and for λ = 0, it is log(y). Box and Cox modified this to handle discontinuities at λ = 0, and later proposed a shifted power transformation to accommodate negative observations. Other variations include the modulus transformation, which normalizes distributions with some symmetry, and the exponential transformation, which can handle skewed data. Estimation of λ is typically done using maximum likelihood or Bayesian methods. Various robust adaptations have been proposed to handle outliers and heteroscedasticity. The Box–Cox transformation has been applied in econometrics, agriculture, and other fields to improve the normality and homoscedasticity of data, facilitating more accurate statistical inference. The transformation has been used to model functional relationships between variables, with the Box–Cox form being widely accepted in econometric studies. However, it is important to note that the transformation does not always satisfy the assumptions of linearity, normality, and homoscedasticity simultaneously. The transformation can also be used to correct for heteroscedasticity and autocorrelation in econometric models. The Box–Cox transformation has been applied in various contexts, including forecasting, prediction, and the analysis of mixed models. However, there are challenges in estimating the transformation parameter, particularly in the presence of outliers or when the data is skewed. The transformation can also introduce bias in the estimation of parameters and affect the interpretation of regression coefficients. In conclusion, the Box–Cox transformation is a powerful tool for improving the normality and homoscedasticity of data, but its effectiveness depends on the specific context and the assumptions of the model. It has been widely used in empirical research, particularly in econometrics, to model functional relationships and improve the accuracy of statistical inference.The Box–Cox transformation is a parametric power transformation technique introduced by Box and Cox (1964) to address issues such as non-additivity, non-normality, and heteroscedasticity in data. This review summarizes the development, applications, and variations of the Box–Cox transformation, as well as its use in statistical modeling and inference. The transformation is defined as a function of the data that depends on a parameter λ, which is estimated to best fit the data. For λ ≠ 0, the transformation is y^λ, and for λ = 0, it is log(y). Box and Cox modified this to handle discontinuities at λ = 0, and later proposed a shifted power transformation to accommodate negative observations. Other variations include the modulus transformation, which normalizes distributions with some symmetry, and the exponential transformation, which can handle skewed data. Estimation of λ is typically done using maximum likelihood or Bayesian methods. Various robust adaptations have been proposed to handle outliers and heteroscedasticity. The Box–Cox transformation has been applied in econometrics, agriculture, and other fields to improve the normality and homoscedasticity of data, facilitating more accurate statistical inference. The transformation has been used to model functional relationships between variables, with the Box–Cox form being widely accepted in econometric studies. However, it is important to note that the transformation does not always satisfy the assumptions of linearity, normality, and homoscedasticity simultaneously. The transformation can also be used to correct for heteroscedasticity and autocorrelation in econometric models. The Box–Cox transformation has been applied in various contexts, including forecasting, prediction, and the analysis of mixed models. However, there are challenges in estimating the transformation parameter, particularly in the presence of outliers or when the data is skewed. The transformation can also introduce bias in the estimation of parameters and affect the interpretation of regression coefficients. In conclusion, the Box–Cox transformation is a powerful tool for improving the normality and homoscedasticity of data, but its effectiveness depends on the specific context and the assumptions of the model. It has been widely used in empirical research, particularly in econometrics, to model functional relationships and improve the accuracy of statistical inference.