Psychiatric disorders such as schizophrenia are commonly accompanied by cognitive impairments that are treatment resistant and crucial to functional outcome. There has been great interest in studying cognitive measures as endophenotypes for psychiatric disorders, with the hope that their genetic basis will be clearer. To investigate this, we performed a genome-wide association study involving 11 cognitive phenotypes from the Cambridge Neuropsychological Test Automated Battery. We showed these measures to be heritable by comparing the correlation in 100 monozygotic and 100 dizygotic twin pairs. The full battery was tested in approximately 750 subjects, and for spatial and verbal recognition memory, we investigated a further 500 individuals to search for smaller genetic effects. We were unable to find any genome-wide significant associations with either SNPs or common copy number variants. Nor could we formally replicate any polymorphism that has been previously associated with cognition, although we found a weak signal of lower than expected P-values for variants in a set of 10 candidate genes. We additionally investigated SNPs in genomic loci that have been shown to harbor rare variants that associate with neuropsychiatric disorders, to see if they showed any suggestion of association when considered as a separate set. Only NRXN1 showed evidence of significant association with cognition. These results suggest that common genetic variation does not strongly influence cognition in healthy subjects and that cognitive measures do not represent a more tractable genetic trait than clinical endpoints such as schizophrenia. We discuss a possible role for rare variation in cognitive genomics.