We prove that if X is a separable, reflexive space which is asymptotic lp for some 1 p , then X embeds into a reflexive space Z having an asymptotic lp finite-dimensional decomposition (FDD). This result leads to an intrinsic characterization of subspaces of spaces with an asymptotic lp FDD. More general results of this type are also obtained. As a consequence, we prove the existence of universal spaces for certain classes of separable, reflexive and asymptotic lp spaces. 2007. Published by Oxford University Press. All rights reserved.