Motivated by problems of vibration in commercial aircraft that caused fatigue in materials, Birnbaum and Saunders (1969a,b) proposed a family of two-parameter distributions to model failure time due to fatigue under cyclic loading and the assumption that failure follows from the development and growth of a dominant crack. This article is cited in at least 930 scientific articles. After the work of Birnbaum and Saunders (1969a,b), the Birnbaum–Saunders model is one of the most important and used models for analyzing lifetime data. Regression analysis is commonly used when covariates are involved in the life test. In this article, we provide a comprehensive review of Birnbaum–Saunders regression models based on parameterizations of the Birnbaum–Saunders distribution in terms of central tendency measures. In particular, we propose a new and straightforward reparameterization of the Birnbaum–Saunders distribution based on an approximation to its mode. In contrast to the original parameterization, the proposed parameterization leads to regression coefficients directly associated (approximately) with the mode of the response variable. We discuss the estimation of model parameters by maximum likelihood, hypothesis testing, and residual analysis. Furthermore, we show that the proposed regression model and other parameterized Birnbaum–Saunders models have the same log-likelihood function estimate when no covariates are used to model the shape parameter. Additionally, we propose a simple script in the R software to solve numerical problems in the Fisher information matrix for some values of the shape parameter. Finally, we present and explore two empirical applications to compare the investigated regression models. © 2022 John Wiley & Sons, Ltd.