General cognitive function is substantially heritable across the human life course from adolescence to old age. We investigated the genetic contribution to variation in this important, health- and well-being-related trait in middle-aged and older adults. We conducted a meta-analysis of genome-wide association studies of 31 cohorts (N=53 949) in which the participants had undertaken multiple, diverse cognitive tests. A general cognitive function phenotype was tested for, and created in each cohort by principal component analysis. We report 13 genome-wide significant single-nucleotide polymorphism (SNP) associations in three genomic regions, 6q16.1, 14q12 and 19q13.32 (best SNP and closest gene, respectively: rs10457441, P=3.93 × 10⁻⁹, MIR2113; rs17522122, P=2.55 × 10⁻⁸, AKAP6; rs10119, P=5.67 × 10⁻⁹, APOE/TOMM40). We report one gene-based significant association with the HMGN1 gene located on chromosome 21 (P=1 × 10⁻⁶). These genes have previously been associated with neuropsychiatric phenotypes. Meta-analysis results are consistent with a polygenic model of inheritance. To estimate SNP-based heritability, the genome-wide complex trait analysis procedure was applied to two large cohorts, the Atherosclerosis Risk in Communities Study (N=6617) and the Health and Retirement Study (N=5976). The proportion of phenotypic variation accounted for by all genotyped common SNPs was 29% (s.e.=5%) and 28% (s.e.=7%), respectively. Using polygenic prediction analysis, ~1.2% of the variance in general cognitive function was predicted in the Generation Scotland cohort (N=5487; P=1.5 × 10⁻¹⁷). In hypothesis-driven tests, there was significant association between general cognitive function and four genes previously associated with Alzheimer's disease: TOMM40, APOE, ABCG1 and MEF2C.