SUMMARY Spatial methods for the analysis of field-plot experiments are considered. Estimators of treatment contrasts obtained by these methods are generally complicated nonlinear functions of the observations. Sufficient conditions for these estimators to be unbiased under a postulated true model are obtained. The analysis of data from field-plot experiments or other spatial experiments generally includes the estimation of treatment contrasts. A common approach to their estimation is to apply ordinary least-squares to a linear model in which the expected value of each observation is taken to be the sum of an overall mean, a treatment effect, and various environmental effects such as block or row and column effects. This approach yields estimators which are linear functions of the observations and which are unbiased if the assumed model is correct. Recently, numerous 'spatial' methods for the analysis of data from field-plot experi- ments have been proposed; see, for example, Bartlett (1978), Wilkinson et al. (1983), Besag & Kempton (1986), Williams (1986) and Gleeson & Cullis (1987). These alternatives to a traditional least-squares analysis attempt to neutralize the disruptive effect that the spatial heterogeneity of the experimental units can have on the estimation of treatment contrasts. Randomization studies of uniformity trial data conducted by Besag & Kempton (1986) indicate that spatial methods yield estimators of treatments contrasts which, in the presence of appreciable spatial heterogeneity, are likely to be more efficient than traditional estimators. However, because the estimators produced by spatial methods are complicated nonlinear functions of the observations, little else is known about their properties. In what follows, we give conditions under which these estimators are unbiased under a postulated true model.