Major depressive disorder (MDD) and Alzheimer's disease (AD) are both common in older age and frequently co-occur. Numerous phenotypic studies based on clinical diagnoses suggest that a history of depression increases risk of subsequent AD, although the basis of this relationship is uncertain. Both illnesses are polygenic, and shared genetic risk factors could explain some of the observed association. We used genotype data to test whether MDD and AD have an overlapping polygenic architecture in two large population-based cohorts, Generation Scotland's Scottish Family Health Study (GS:SFHS; N=19 889) and UK Biobank (N=25 118), and whether age of depression onset influences any relationship. Using two complementary techniques, we found no evidence that the disorders are influenced by common genetic variants. Using linkage disequilibrium score regression with genome-wide association study (GWAS) summary statistics from the International Genomics of Alzheimer's Project, we report no significant genetic correlation between AD and MDD (rG=-0.103, P=0.59). Polygenic risk scores (PRS) generated using summary data from International Genomics of Alzheimer's Project (IGAP) and the Psychiatric Genomics Consortium were used to assess potential pleiotropy between the disorders. PRS for MDD were nominally associated with participant-recalled AD family history in GS:SFHS, although this association did not survive multiple comparison testing. AD PRS were not associated with depression status or late-onset depression, and a survival analysis showed no association between age of depression onset and genetic risk for AD. This study found no evidence to support a common polygenic structure for AD and MDD, suggesting that the comorbidity of these disorders is not explained by common genetic variants.