Cognitive impairment is a major determinant of functional outcomes in schizophrenia, and efforts to understand the biological basis of cognitive dysfunction in the disorder are ongoing. Previous studies have suggested genetic overlap between global cognitive ability and schizophrenia, but further work is needed to delineate the shared genetic architecture. Here, we apply genomic structural equation modelling to identify latent cognitive factors capturing genetic liabilities to 12 cognitive traits measured in the UK Biobank (UKB). We explore the overlap between latent cognitive factors, schizophrenia, and schizophrenia symptom dimensions using a complementary set of statistical approaches, applied to data from the latest schizophrenia genome-wide association study (Ncase = 53,386, Ncontrol = 77,258) and the Thematically Organised Psychosis study (Ncase = 306, Ncontrol = 1060). We identified three broad factors (visuo-spatial, verbal analytic and decision/reaction time) that underly the genetic correlations between the UKB cognitive tests. Global genetic correlations showed a significant but moderate negative genetic correlation between each cognitive factor and schizophrenia. Local genetic correlations implicated unique genomic regions underlying the overlap between schizophrenia and each cognitive factor. We found evidence of substantial polygenic overlap between each cognitive factor and schizophrenia but show that most loci shared between the latent cognitive factors and schizophrenia have unique patterns of association with the cognitive factors. Biological annotation of the shared loci implicated gene-sets related to neurodevelopment and neuronal function. Lastly, we find that the common genetic determinants of the latent cognitive factors are not predictive of schizophrenia symptom dimensions. Overall, these findings inform our understanding of cognitive function in schizophrenia by demonstrating important differences in the shared genetic architecture of schizophrenia and cognitive abilities.