The writing and publication of unfinished texts has long been a recognised feature of the British Romantic period. Recent scholarship has observed that the reading public in the late 18th century and early 19th century became accustomed to the fragment, and found it acceptable and even fashionable: the idea of an unfinished text evolved from being a failure of genre, into a quasi-genre in it own right. Poets such as Coleridge, Byron, Shelley and Keats published texts in unfinished forms, and often advertised this fact rather than apologised for it. One established critical method of reading the Romantic fragment poem is in relation to key features of German Romantic philosophy, especially the fragmentary writings of Friedrich Schlegel, who developed a theory of the artistic or philosophical fragment as a radiant moment that reached beyond its own boundaries. Modern interpretations of Romantic fragments have included formalist, deconstructionist and New Historicist approaches. This article provides a survey of the most influential overviews of the Romantic fragment, arguing for the importance of maintaining a simple common terminology that distinguishes between those fragments that were deliberately published by the author, and those that are recovered by editors. There is some analysis of examples from Coleridge, Ann Batten Cristall and Lord Byron, in which the versatility of fragment poems is explored, with an emphasis on self-reflexivity, the text providing an allegory of its own reading. The article argues that fragment poems are highly useful in the teaching of Romanticism, since they demand bold, imaginative readings that project a resolution beyond the text, and thus dramatise in miniature some key aspects of Romantic thought and aesthetics, such as thwarted idealism, and the visionary longing for the absolute.