Prior to this study, the prevalence of obstructive sleep apnoea/hypopnoea syndrome (OSAHS) in adults with Down syndrome was unknown. We hypothesized that unrecognised OSAHS could have an additional deleterious impact on mood and behavioural disturbances in this group of people. Aims: To assess the prevalence of OSAHS in adults with Down syndrome in the United Kingdom, subjectively and objectively, and ascertain its association with diurnal behavioural disturbances. Method: Cross‐sectional questionnaire study with home sleep apnoea testing (HSAT) during 2011–2015 across the four nations of the United Kingdom. Participants were adults aged ≥16 years with Down syndrome. Main outcome measures were: self‐ or caregiver‐completed questionnaire data, including the Pictorial Epworth Sleepiness Scale (pESS), selected domains of the Developmental Behavioural Checklist for Adults (DBC‐A), anthropometric measures, and symptoms of OSAHS. Objective prevalence was undertaken in a sample of responders using HSAT. Results: Responses were received from 1321/5270 participants (25%), with 1105 valid responses (21%). Eighty‐one participants (7%) reported a prior diagnosis of OSA, of whom 38 were receiving therapy. Using validated algorithms, a diagnosis of OSAHS was probable in 366 participants (35%), who were younger, with higher BMI and higher mean total pESS (p < 0.0001). A total of 23% of participants had a pESS > 10. OSAHS was a strong marker for behavioural disturbances on the DBC‐A depression, disruption and anti‐social subscales (p < 0.001). Of 149 individuals who underwent HSAT, 42% were diagnosed with OSAHS. Conclusions: Untreated OSAHS in Down syndrome is common and associated with behavioural and mood disturbances. Improving awareness of OSAHS amongst adults with Down syndrome, their families and healthcare professionals is essential.