The progress in animal improvement depends on the accuracy with which the breeding value of each individual in the breed, or herd, is estimated, and how the animals chosen on the basis of our estimates are combined in matings. We are concerned here only with the first part of the problem, i.e. the estimation of breeding values. The estimates are made in regard to certain characters, or traits, which are of particular interest from an economic point of view, and they may be based on the phenotypic merit of the individual, or on the merits of its ancestors or collateral relatives, or on the merits of its progeny, in regard to the character in question. Often a combination of two, or three, or all four methods may be used. Their relative importance depends on the heritability of the character, as will be discussed later.