Quadratic estimates of variance components for various linear models were investigated;For the general balanced nested classification with no specific distributions assumed, it was shown that the quadratic estimate which was unbiased and which had minimum variance was given by the analysis of variance method of estimating the variance components;For the one fold classification with unequal numbers the model is yij=m+ai+bij where E ai2 = sigma a2; E bij2 = sigma 2;The best quadratic unbiased estimate of sigmaa2 depends on sigma2 and sigmaa2. Under the above model with the additional assumptions E ai 4 = 3 sigmaa4 and E bij 4 = 3 sigma4 the quadratic estimation of sigma a2 was investigated. Two methods of estimation were given, and a method of determining a lower bound on the efficiency of each method was given if w = sigmaa2/sigma2 is known. It is believed in most cases when w > .01, that one of these two methods will give an efficiency above 80 percent;For the general balanced cross classification with normality assumptions it was shown that the best quadratic unbiased estimate of the variance components is given by the analysis of variance method of estimating variance components. It was also shown that the best quadratic unbiased estimate of any linear combination of variance components is given by the same linear combination of the best quadratic unbiased estimates of the individual variance components.