Within the last 20 years, the US has mounted amassive campaign against invasions bynon-indigenous species (NIS) such as zebramussels, kudzu, water hyacinths, and brown treesnakes. NIS have disrupted native ecosystemsand caused hundreds of billions of dollars ofannual damage. Many in the scientificcommunity say the problem of NIS is primarilypolitical and economic: getting governments toregulate powerful vested interests thatintroduce species through such vehicles asships' ballast water. This paper argues that,although politics and economics play a role,the problem is primarily one of scientificmethod. Even if commercial interests werewilling to spend the necessary funds to controlNIS, and even if government were willing toregulate them, ecological theory is notadequate to provide clear direction for eithereffort. The paper argues there is nocomprehensive, predictive “theory ofinvasibility,” as part of a larger theory ofcommunity structure, that might guideecological decision making regarding NIS, andfor at least three reasons. (1) There is nofirm definition of “NIS,” “native,” “exotic,”and so on, and ecologists do not use the termsconsistently; as a result, biologists debatingvarious accounts of community structure andecological explanation often do not even makelogical contact with each other. (2) Thedominant theory used to understandinvasibility, island biogeography, has noprecise predictive power and is unable toclarify when NIS might promote biodiversity andwhen they might hinder it. (3) There are nofirm, empirical generalizations that revealwhen a colonizer or a NIS might be likely totake over a new environment, and when it mightnot succeed in doing so. As a result,scientists have only rough “rules of thumb” toshore up their arguments against NIS. Given theincompleteness of current ecological theory,the paper closes with several suggestions forways that study of NIS might enhanceunderstanding of basic commmunity structuresand vice versa.