Linear mixed effects models are widely used in practice because they are flexible and can handle a broad range of data types. A key part in the analysis of data is the use of diagnostic methods to assess the plausibility of a fitted model. In regression modelling, graphical diagnostic methods make use of residuals that are defined straightforwardly. The dependence structure and the presence of multiple error terms in linear mixed models makes it more difficult to define residuals for linear mixed models. This project will involve reviewing some of the proposals for defining residuals and exploring some of their properties.