A considerable number of individuals infected with SARS-CoV-2 continue to experience symptoms after the acute phase. Here, we report findings from a nationwide questionnaire study in Denmark including 61,002 RT-PCR confirmed SARS-CoV-2 cases and 91,878 test-negative controls aged 15-years or older. Six to twelve months after the test, the risks of 18 out of 21 symptoms were elevated among test-positives. The largest adjusted risk differences (RD) were observed for dysosmia (RD = 10.92%, 95% CI 10.68–11.21%), dysgeusia (RD = 8.68%, 95% CI 8.43–8.93%), fatigue/exhaustion (RD = 8.43%, 95%CI 8.14–8.74%), dyspnea (RD = 4.87%, 95% CI 4.65–5.09%) and reduced strength in arms/legs (RD = 4.68%, 95% CI 4.45–4.89%). During the period from the test and until completion of the questionnaire, new diagnoses of anxiety (RD = 1.15%, 95% CI 0.95–1.34%) or depression (RD = 1.00%, 95% CI 0.81–1.19%) were also more common among test-positives. Even in a population where the majority of test-positives were not hospitalized, a considerable proportion experiences symptoms up to 12 months after infection. Being female or middle-aged increases risks.