SUMMARY Definitions of robust maximum likelihood (robust ML) and robust restricted maximum likelihood (robust REML) are introduced, and the definitions are applied to data from biological and chemical experiments. A simulation study is undertaken to investigate the asymptotic properties of robust ML and robust REML in small samples and to examine the advantages of using robust methods. 1. Introduction and Definitions Linear models with multiple sources of error are widely used in designed experiments across many scientific fields. An example of such an experiment is described by Patterson and Nabugoomu (1992), from Patterson and Silvey (1980). Six varieties of wheat were grown at ten centres that formed a sample of the main types of growing area for wheat in Scotland, and the yields in tonnes/hectare were recorded. The experiment is unbalanced because, of 60 possible variety-centre combinations, only 46 were used. At seven centres, four varieties were grown and at the remaining three centres, all six varieties were grown. In this paper we fit the simplest mixed linear model proposed for this data by Patterson and Nabugoomu, namely: