Over the past several years, we have been studying the development of cognitive abilities in infants, differences between preterms and full-terms on these infant abilities, and the relation o f infant abilities to later outcome. In the course o f this work, we have also probed the specificity o f the later cognitive deficits associat­ ed with prematurity. In the present chapter, we review some of our findings, fo­ cusing on: (a) cognitive differences between preterms and full-terms in infancy, (b) cognitive differences between them in later childhood, (c) the relation of in­ fant to childhood abilities, (d) the role o f memory and processing speed, two spe­ cific abilities, in explaining preterm deficits on more global measures of cogni­ tion, and finally, (e) the relation of Respiratory Distress Syndrome (RDS), a medical risk factor prevalent in preterms, to memory impairments in infancy and later childhood.