Introgression, the transfer of genetic material between species through hybridisation, occurs in many taxa and has important consequences. Genomic studies allow us to characterise the landscape of introgression across the genome, shedding light on both its adaptive benefits and the incompatibilities that help to maintain species barriers. Studies taking a genome-wide view suggest that adaptive introgression may be common, but that introgressed variation between many species is selected against throughout much of the genome. Confounding factors can complicate interpretations from these data, and computational simulations have proved vital to illustrate expected patterns under different scenarios. Future developments will move beyond correlative evidence to explicit models that account for how selection and genetic drift influence introgressed variation.