This is an open-label study of the combination of atomoxetine and oxybutynin (ato-oxy) in children with Down syndrome and obstructive sleep apnea (OSA) documented by polysomnography (PSG). Participants will receive ato-oxy for 6 months. Ato-oxy dose will be 5 mg oxybutynin and 0.5mg/kg/day (max 40 mg) atomoxetine. Dosing of the study treatment will occur approximately 30 minutes prior to bedtime. Participants who withdraw from the study will not be replaced. Study participants will undergo eligibility screening that will include an initial screening to determine whether non- PSG enrollment criteria are met, followed by a 1 night in-lab PSG and health-related quality of life (HRQOL) and cognitive assessment for participants who qualify based on non-PSG criteria. For participants who are eligible and enroll in the study, the screening PSG night will serve as the baseline measure for apnea hypopnea index (AHI) and other PSG endpoints. On the final night of dosing for ato-oxy participants will return for inpatient PSG and health-related quality of life assessment and cognitive assessment. The primary efficacy endpoint is the change in obstructive AHI from baseline.