Explicit solutions have been derived for the maximum likelihood (ML) and restricted maximum likelihood (REML) equations under normality for four common variance components models with balanced (equal subclass numbers) data. Solutions of the REML equations are identical to analysis of variance (AOV) estimators. Ratios of mean squared errors of REML and ML solutions have also been derived. Unbalanced (unequal subclass numbers) data have been used in a series of numerical trials to compare ML and REML procedures with three other estimation methods using a two-way crossed classification mixed model with no interaction and zero or one observation per cell. Results are similar to those reported by Hocking and Kutner [1975] for the balanced incomplete block design. Collectively, these studies and those of Klotz, Milton and Zacks [1969] point, with some exceptions, to the greater efficiency of ML estimators under a range of experimental settings.