Residual maximum likelihood estimation is a method used in linear models to estimate covariance parameters. It considers the loss of degrees of freedom in estimating the mean and provides unbiased estimating equations for the variance parameters. The approach in this paper shows that REML has an exact conditional likelihood interpretation, which removes dependence on nuisance parameters. This interpretation clarifies the motivation for REML and can be applied to non-normal models with low-dimensional sufficient statistics. The researchers extended the concept of REML to generalized linear models with varying dispersion and canonical link, providing explicit calculations for the one-way layout and a saddlepoint approximation for the conditional likelihood.