Linear B is the earliest form of writing used for Greek. The tablets written in this script offer crucial information about the Mycenaean Greeks and their time. This Companion aims at not only summarizing the results of current research but also trying to explain the problems which arise from the study of the texts and the methods which can be used to solve them. It is aimed both at the beginner who needs an introduction to this area and to advanced scholars (archaeologists, historians, classicists) who require an up-to-date account which can serve as a standard reference tool and highlight the remaining problems.