The incidence of apneic spells during different sleep states, active sleep, quiet sleep, and undifferentiated sleep was determined in eight preterm infants of 30 to 35 weeks' conceptional age, by means of a polygraphic recording technique. They were free of perinatal and postnatal complications other than apnea. During their active or rapid eye movement (REM) sleep they showed significantly more apneic episodes which were also longer lasting and they were accompanied by bradycardia of a greater severity. The organization of the immature nervous system with a preponderance of inhibitory synaptic connections and the additional inhibition of spinal motoneurons during REM sleep are likely to be the cause of apneic spells in otherwise "normal" preterm infants.