1. Distributions of surnames were examined using data from about 40000 individuals from consanguineous marriages (1930-59) in the island of Sardinia. They fit the Karlin-McGregor (1967) distribution for neutral alleles.

2. The logarithmic distribution by Fisher (1943) and the Karlin-McGregor (1967) distribution give practically indistinguishable fits to these data.

3. The two parameters underlying the Fisher distribution, a and v, are fully interdependent, given the sample size N. From the latter and from the number of surnames S, they can be easily and satisfactorily estimated.

4. The quantity a measures immigration to the area and can be taken as a measure of the richness of the gene pool, being closely related to the quantity θ=Neμ (Ne= effective population size, μ= mutation rate) after the necessary corrections have been taken care of.

5. Surnames behave like genes transmitted by the male line. Differences of female and male migrations require correction for comparison with data from autosomal gene frequencies but there seem to be no important differences between female and male migration in this area, as judged by analysis of pedigrees of consanguineous marriages.

6. It is probable that the migration estimates obtained in this material are lower than real ones, mostly because of a bias characteristic of the particular source of surnames here employed.

7. The distribution of surnames from areas which have very recently undergone economic development is not at equilibrium and is not fitted as well as that of areas which have had less important recent changes. As might be expected, the disturbance is particularly marked in the class most sensitive to effects of recent increases in migration, that of rarest surnames (represented by only one individual). In fact in this material it is noticeable only in this class and correction can be made for it.

8. At equilibrium of migration and drift, the number of surnames in a population sample of given size can give a complete description of the population structure for that sample, in the same sense that the number of alleles is a sufficient statistic for the study of neutral evolution. The study of surnames, given their nature of ‘neutral’ alleles, the large number of alleles and the ease with which large numbers of individual data can be collected, can be a valuable help in the study of genetic population structure.