The article provides a detailed description of the `lme4` package for R, which is used to fit linear mixed-effects models. It covers the formulation and representation of linear mixed models, including the structure of the model, the evaluation of the profiled deviance or REML criterion, and the structure of classes representing such models. The article also discusses the modular structure of the `lmer` function, which is composed of four main modules: formula parsing, objective function construction, optimization, and output interpretation. The authors provide examples and explanations of mixed-model formulas, the algebraic and computational aspects of random effects, and the penalized least squares method used for maximum likelihood estimation. The article aims to provide a comprehensive guide for users who wish to fit specialized linear mixed models using the `lme4` package.The article provides a detailed description of the `lme4` package for R, which is used to fit linear mixed-effects models. It covers the formulation and representation of linear mixed models, including the structure of the model, the evaluation of the profiled deviance or REML criterion, and the structure of classes representing such models. The article also discusses the modular structure of the `lmer` function, which is composed of four main modules: formula parsing, objective function construction, optimization, and output interpretation. The authors provide examples and explanations of mixed-model formulas, the algebraic and computational aspects of random effects, and the penalized least squares method used for maximum likelihood estimation. The article aims to provide a comprehensive guide for users who wish to fit specialized linear mixed models using the `lme4` package.