FST is a fundamental measure of genetic differentiation and population structure currently defined for subdivided populations. FST in practice typically assumes the "island model", where subpopulations have evolved independently from their last common ancestral population. In this work, we generalize the FST definition to arbitrary population structures, where individuals may be related in arbitrary ways. Our definitions are built on identity-by-descent (IBD) probabilities that relate individuals through inbreeding and kinship coefficients. We generalize FST as the mean inbreeding coefficient of the individuals' local populations relative to their last common ancestral population. This FST naturally yields a useful pairwise FST between individuals. We show that our generalized definition agrees with Wright's original and the island model definitions as special cases. We define a novel coancestry model based on "individual-specific allele frequencies" and prove that its parameters correspond to probabilistic kinship coefficients. Lastly, we study and extend the Pritchard-Stephens-Donnelly admixture model in the context of our coancestry model and calculate its FST. Our probabilistic framework provides a theoretical foundation that extends FST in terms of inbreeding and kinship coefficients to arbitrary population structures.