The distribution of species in ecosystems is influenced by several biotic and abiotic factors, and physical barriers operate as filters to dispersion. The ancient processes of human migration and settlement, and the recent globalization trends are the main responsible factors for the transposition of geographic barriers by organisms and species introduction in regions outside their original distribution range. In order to designate introduced species, the adopted terminology may eventually lead to a lack of operational definitions, misleading the interpretation of paradigms concerning the invasion process. According to the literature, the criteria i) biogeographic, ii) population growth, and iii) spreading in the invaded region, should be taken into account as a manner to avoid subjective interpretations. Most of the presented models are inserted in the same paradigms to explain the status of the invasion process (arrival, establishment and invasion), although each model has its own concepts. Many authors consider both characteristics of the invader species and of the invaded environments to explain species success and establishment. The use of clear concepts, well established and non subjective concepts of introduced and invading species as well as its derivatives is crucial to unify the paradigms of the invasion process.