Genome introgressions drive evolution across the animal, plant and fungal kingdoms. We use the Saccharomyces species complex to investigate the birth, evolution and function of interspecies introgressions. We developed a computational framework to map shared alleles across thousands of Saccharomyces genomes. We detected multiple pulses of introgressions by reconstructing the major hybridisation events and unveiled a common role of genome instability in initiating introgressions. The large majority of segregating introgression blocks derive from recent hybridisation and are clade-specific, while a handful of introgressions are very ancient and shared across many strains. We used quantitative genetics to probe the fitness landscape of introgressions across environments and life-cycle phases. We detected few cases of adaptive introgressions that modulate asexual growth or sexual reproduction, while the large majority appear to be neutral. Population genomics coupled with experiments enable us to retrace history, emergence and functional implications of gene flow across species boundaries.