This narrative review provides a broad perspective on immature control of breathing, which is universal in infants born premature. The degree of immaturity and severity of clinical symptoms are inversely correlated with gestational age. This immaturity presents as prolonged apneas with associated bradycardia or desaturation, or brief respiratory pauses, periodic breathing, and intermittent hypoxia. These manifestations are encompassed within the clinical diagnosis of apnea of prematurity, but there is no consensus on minimum criteria required for diagnosis. Common treatment strategies include caffeine and noninvasive respiratory support, but other therapies have also been advocated with varying effectiveness. There is considerable variability in when and how to initiate and discontinue treatment. There are significant knowledge gaps regarding effective strategies to quantify the severity of clinical manifestations of immature breathing, which prevent us from better understanding the long-term potential adverse outcomes, including neurodevelopment and sudden unexpected infant death.