Arm A will include continued inpatient monitoring of apnea, bradycardia, and/or desaturation until an event free period of time (5 days per the current standard of care at each participating institution) has been established and deemed appropriate for discharge home per the discretion of the responsible provider. The protocol described in Arm A is currently utilized as a standard of care for many infants locally and nationally and is the first management option utilized at all participating institutions. Arm B will include the initiation of caffeine treatment once randomization has been established. Per current dosing guideline, patients will receive a one-time loading dose of 20 milligrams per kilogram on day 1 with a daily maintenance dose thereafter of 10 milligrams per kilogram. Monitoring of caffeine drug levels will not be included in our protocol as the safety of caffeine treatment in neonates without laboratory monitoring has been well established. After receiving the loading dose of caffeine, infants will receive continued inpatient monitoring of apnea/bradycardia/desaturation until an event free period of time (3 days per the current standard of care at each participating institution) has been established and deemed appropriate for discharge home per the discretion of the responsible provider. Infants discharged home on caffeine therapy will receive instructions regarding use of a home monitor, with alarms set to alarm for heart rate <80 or > 200 beats per minute, or for oxygen saturation level < 90%. The protocol described in Arm B is currently utilized as a standard of care for many infants locally and nationally as a secondary option after observation alone has resulted in a prolonged inpatient stay due to persistent apnea with all other discharge criteria having been met. In a recent retrospective analysis of home monitor use, the investigators found that 1 in 20 infants with apnea of prematurity were discharge home with a monitor.