The past three decades have seen a number of monographs and articles on the phonetics of Icelandic, but little has yet been done with its phonemics. A first attempt was made by Kemp Malone in a paper read to the Linguistic Society in 1950, but this was drastically amended in a note he published three years later in LANGUAGE. There is a vigorous dogmatism in Malone's approach to phonemics, unhampered by those ifs and buts which strew the discourse of more cautious scholars. Although he claims to have taken account of phonemic work done since 1923, when he published his Phonology of Modern Icelandic, there are no references to indicate the scope of his reading, and few traces in his work of the ideas advanced by other scholars. His only statement of principle is this: 'I have tried first of all to classify each isolable speech-sound as an allophone of some phoneme.' He provides minimal pairs for each phoneme and a brief statement of its distribution. But we do not learn just how he isolated these speech sounds and overlooked others, or what criteria permitted him to classify the sounds with one phoneme and not another. Neither is there any description of the larger units into which the phonemes enter, or any consideration of alternative analyses such as we have been taught to expect in recent phonemic work. That alternatives are possible becomes apparent in his reconsideration of the prosodic system, which permitted him to reduce the number of phonemes at one blow from 61 to 39. This change suggests that there is room for further consideration of Icelandic phonemics. It will be the purpose of this paper to explore various possible analyses of Modern Icelandic as a contribution to what has been called the 'indeterminacy' of phonemic systems and the ‘convertibility’ of different descriptions. We shall also take up some of the problems involved in finding the structure or structures of a language.