Anxiety is common in children with ASD; however, the burden of specific anxiety disorders for adults with ASD is under-researched. Using the Stockholm Youth Cohort, we compared anxiety disorder diagnoses among autistic adults (n = 4049), with or without intellectual disability, and population controls (n = 217,645). We conducted additional sibling analyses. Anxiety disorders were diagnosed in 20.1% of adults with ASD compared with 8.7% of controls (RR = 2.62 95% CI 2.47–2.79), with greatest risk for autistic people without intellectual disability. Rates of almost all individual anxiety disorders were raised, notably obsessive–compulsive disorder and phobic anxiety disorders. Anxiety disorders were more common in full siblings and half-siblings of people with ASD. The implications of this are explored.