The inature of variability in a populationi observed at a giveni time wTith respect to a particular metric trait is of great interest and importance among animal and plant breeders. Their work depends heavily upon the ability to design breeding experiments and to take advantage of statistical techniques which will enable them successfully to apportion differences in such a trait to the various broad causal factors operating upon the individuals constituting the population. This they must accomplish with sufficient accuracy to describe to some extent the genetic and environmental complex affecting the trait and to predict breeding results. It has been shown, particularly by the work of Fisher, Haldane and Wright, that for various quantitative traits the system of genes involved does have average properties which are measurable and the analysis of variance has proved to be a powerful tool in the estimation of such parameters. This paper is presented as a review of some of the applications of variance componeints in statistical genetics and of some statistical problems commonly encountered in their use in this field. The situation frequently to be met in quantitative genetics is as follows: we have a set of data arranged in a particular type of classification and described by a linear function of effects of various classes and subclasses. Generally this model is that which Eisenhart (1947) has called Model II, in which all elements except A are regarded as random variables, although it may frequently be what he called the Mixed Model, in which certain of the effects are regarded as fixed rather than as random variables. The first step then is the estimation of the variances of these random variables and the second step the linear combination of certain of these estimates to provide further estimates of the parameters of heredity, by which I mean any of the parameters, genetic and environmental, describing the variability of the quantitative trait. Weinberg (1910) showed that the correlation between parent and offspring is 1/2 o-/Io in a random breeding population, where ois