The connections between the genetic similarity of pairs of individuals and their phenotypic similarity has been at the heart of quantitative genetics for over a century. Until recently there was in effect only one definition of kinship, based on expected genome sharing from pedigrees. But pedigrees only specify expected genome sharing, whereas nowadays genome-wide genotyping allows genomic similarity between individuals to be measured directly, allowing traditional notions of heritability to be extended to many different uses. The problem is that there are many different similarity measures available, and little recognition of this issue or its implications in the genetics literature, which contributes to confusion about the interpretation of modern heritability analyses. It turns out that defining kinship in a principled and useful way is surprisingly difficult, and it may now be time to abandon the idea that there is a useful single-number, or even low-dimensional, measure of relatedness. I review these ideas and present some new approaches to genetic analyses based on multi-SNP measures of genetic similarity. This is joint work with postdoc Doug Speed, who is funded by the UK Medical Research Council.