It is assumed that host genetic variability for susceptibility to infection will necessarily condition the rates of virus evolution, either by driving them to diversification into strains that track the different host defense alleles (e.g., antigenic diversity), or by canalization to infect only the most susceptible genotypes. Associated to these processes, virulence may or may not increase. To tackle these questions, we performed evolution experiments with turnip mosaic virus (TuMV) in genotypes of Arabidopsis thaliana differing in their susceptibility to infection. TuMV lineages evolved in A. thaliana mutants in genes involved in resistance pathways and in genes whose products are essential for potyviruses infection. Plant genotypes classified into three categories: hypersusceptible (severer symptoms), equal to wild-type and hyposusceptibles (weaker or no symptoms). We found that evolution proceeded faster in hyposusceptible hosts than in hypersusceptible ones. Most of the phenotypic differences shown by the ancestral virus across host genotypes were removed after evolution, suggesting the combined action of selection and drift. When all evolved viral lineages were tested in all plant genotypes used in the experiments, we found compelling evidences that the most restrictive plant genotypes selected for more generalist viruses, while more permissive genotypes selected for more specialist viruses. Sequencing the genomes of the evolved viral lineages, we found that selection targeted the multifunctional protein VPg in most host genotypes.