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Background  The high fecundity of fish species allows intense selection to be practised and therefore leads to fast genetic gains. Based on this, numerous selective breeding programmes have been started in Europe in the last decades, but in general, little is known about how the base populations of breeders have been built. Such knowledge is important because base populations can be created from very few individuals, which can lead to small effective population sizes and associated reductions in genetic variability. In this study, we used genomic information that was recently made available for turbot ( Scophthalmus maximus ), gilthead seabream ( Sparus aurata ), European seabass ( Dicentrarchus labrax ) and common carp ( Cyprinus carpio ) to obtain accurate estimates of the effective size for commercial populations.    Methods  Restriction-site associated DNA sequencing data were used to estimate current and historical effective population sizes. We used a novel method that considers the linkage disequilibrium spectrum for the whole range of genetic distances between all pairs of single nucleotide polymorphisms (SNPs), and thus accounts for potential fluctuations in population size over time.    Results  Our results show that the current effective population size for these populations is small (equal to or less than 50 fish), potentially putting the sustainability of the breeding programmes at risk. We have also detected important drops in effective population size about five to nine generations ago, most likely as a result of domestication and the start of selective breeding programmes for these species in Europe.    Conclusions  Our findings highlight the need to broaden the genetic composition of the base populations from which selection programmes start, and suggest that measures designed to increase effective population size within all farmed populations analysed here should be implemented in order to manage genetic variability and ensure the sustainability of the breeding programmes.    Supplementary Information  The online version contains supplementary material available at 10.1186/s12711-021-00680-9.

Estimates of recent and historical effective population size in turbot, seabream, seabass and carp selective breeding programmes