Rapid adaptation of invasive species to novel habitats has puzzled evolutionary biologists for decades, especially as this often occurs in the face of limited genetic variability. Although some ecological traits common to invasive species have been identified, little is known about the possible genomic/genetic mechanisms that may underlie their success. A common scenario in many introductions is that small founder population sizes will often lead to reduced genetic diversity, but that invading populations experience large environmental perturbations, such as changes in habitat and environmental stress. Although sudden and intense stress is usually considered in a negative context, these perturbations may actually facilitate rapid adaptation by affecting genome structure, organisation and function via interactions with transposable elements (TEs), especially in populations with low genetic diversity. Stress-induced changes in TE activity can alter gene action and can promote structural variation that may facilitate the rapid adaptation observed in new environments. We focus here on the adaptive potential of TEs in relation to invasive species and highlight their role as powerful mutational forces that can rapidly create genetic diversity. We hypothesise that activity of transposable elements can explain rapid adaptation despite low genetic variation (the genetic paradox of invasive species), and provide a framework under which this hypothesis can be tested using recently developed and emerging genomic technologies.This article is protected by copyright. All rights reserved.