More than sixty years after Michael Ventris’ decipherment of the Linear B script, 14 of its 87 syllabic signs still have no sound-values assigned to them. This group of ‘undeciphered’ signs represent a significant gap in our ability not only to read the Linear B script, but also to understand its development and use. Chapter 1 of this thesis analyses the origins and usage of signs with known sound-values to establish what types of values are in principle most likely to be found amongst the undeciphered signs: this investigation also enables an exploration of the development of Linear B, its relationship with its parent script Linear A, and the motivating factors underlying the creation of new Linear B signs. Chapter 2 consists of studies of each individual undeciphered sign, including a palaeographic analysis of their forms, discussion of their corpus of attestations, and an examination of their prospects of decipherment and possible sound-values in the light of the results of Chapter 1. Finally, Chapter 3 employs this group of signs in a case-study to explore the potential of palaeographic analysis to contribute to our understanding of wider issues concerning the Linear B script and its context of use within the Mycenaean palaces. This case-study focuses in particular on two main uses of palaeography: as a means of assigning a relative chronology to Linear B texts, and as evidence for the reconstruction of the Mycenaean scribes’ administrative work and training.