Males and females commonly show pronounced differences in a range of phenotypes, including form, behaviour and physiology. Yet, the mostly shared genome constrains independent evolution in the sexes, generating sexual conflict. The male-limited Y chromosome should be important for mitigating sexual conflict by accumulating sexually antagonistic loci that only benefit males, but the adaptive potential of Y chromosomes is limited by the loss of recombination between the sex chromosomes. We have investigated how sexual body size dimorphism can evolve in a seed beetle Callosobruchus maculatus, and what is the role of sex chromosomes, using a combination of artificial selection, quantitative genetics and long-read sequencing. In this seminar I will show how different forms of sex-specific selection affect the evolutionary potential of size dimorphism, and present our resent findings characterizing genetic variation in the Y chromosome associated with size dimorphism.