Reading and language abilities are critical for educational achievement and success in adulthood. Variation in these traits is highly heritable, but the underlying genetic architecture is largely undiscovered. Genetic studies of reading and language skills traditionally focus on children with developmental disorders; however, much larger unselected adult samples are available, increasing power to identify associations with specific genetic variants of small effect size. We introduce an Australian adult population cohort (41.7–73.2 years of age, N = 1505) in which we obtained data using validated measures of several aspects of reading and language abilities. We performed genetic association analysis for a reading and spelling composite score, nonword reading (assessing phonological processing: a core component in learning to read), phonetic spelling, self-reported reading impairment and nonword repetition (a marker of language ability). Given the limited power in a sample of this size (~80% power to find a minimum effect size of 0.005), we focused on analyzing candidate genes that have been associated with dyslexia and developmental speech and language disorders in prior studies. In gene-based tests, FOXP2, a gene implicated in speech/language disorders, was associated with nonword repetition (p < .001), phonetic spelling (p = .002) and the reading and spelling composite score (p < .001). Gene-set analyses of candidate dyslexia and speech/language disorder genes were not significant. These findings contribute to the assessment of genetic associations in reading and language disorders, crucial for understanding their etiology and informing intervention strategies, and validate the approach of using unselected adult samples for gene discovery in language and reading.