Most, if not all, alien plant species of the family Acanthaceae (acanths) found in tropical islands were intentionally introduced as garden ornamentals, because of their showy coloured flowers, bracts or leaves. Some have ‘escaped’ gardens and have naturalized in human-disturbed areas as weeds, adventives, or ruderal species. A few species have successfully invaded secondary and relatively undisturbed native wet forests. This paper reviews the naturalized alien acanths in tropical islands, and focuses on the currently invasive and potentially invasive species. This study is based on recent (1994–2004) botanical surveys and field observations conducted in several tropical oceanic islands of the Pacific and Indian Oceans, and on bibliographical searches of other tropical islands and countries. A total of 52 acanth species are naturalized in the Indo-Pacific islands; 26 of them are native to Asia, 18 to tropical America, and only 8 to Africa. The number of naturalized acanths in selected tropical oceanic and continental islands varied from 7 to 25 species and the rate of naturalized acanths (number of naturalized species/number of introduced species) from 27% to 62%. We recorded eight major invasive species: the erect herbs or shrubs Justicia carnea, Odontonema strictum, Phlogacanthus turgidus, Sanchezia speciosa and Strobilanthes hamiltonianus form dense monospecific thickets in the understorey of wet forests; the woody vine Thunbergia grandiflora smothers native trees; the creeping herb Hemigraphis alternata forms dense carpets that totally cover the ground; and the herb Ruellia brevifolia colonizes the understorey of closed-canopy wet forest. We also discuss eight potentially or incipient invasive acanths which are subspontaneous or sparingly naturalized, but which are not yet considered invasive. Most of the currently invasive acanths are sterile, the lack of fruit production being explained by the absence of pollinators and/or to their particular floral structure and reproduction modes. They reproduce vegetatively by stem fragmentation or by root suckers, and their range expansion is thus relatively slow. Their success may be attributed to their long (50–100 years) residence times, and long-distance dispersal by humans. Many potentially invasive acanths are newly introduced ornamentals that produce seeds, thus constituting potentially greater threats. The Acanthaceae is not yet recognized as an ‘aggressive’ plant family (e.g. compared to the Fabaceae, Melastomataceae, Poaceae, or Rosaceae). It is, however, one of the most popular ornamental families in the Tropics, and should receive more attention because of the increasing number of current and incipient invasive species found in tropical islands.