Centrifugal compressors are widely used in different fields. Their design requires high performance and a wide operating range, where, at lower mass flow rates, unstable flow dynamic phenomena occur, which are extremely harmful and, at the same time, complex to fully understand. This review paper presents the main research from the last 40 years on the subject of instability in centrifugal compressors, aiming to clarify the main (sometimes contradictory) causes, classifying them according to the component in which they are triggered or the interaction between them. Importance is given to works that develop criteria for the identification of the stability limit with simplified models. The main techniques used to extend the stability limit are also presented by distinguishing between passive and active fixed-flow control methods; moreover, the main works on variable geometry techniques are reported, showing the advantages and disadvantages of their use. Finally, an overview of the innovative applications of centrifugal compressors, such as fuel cells, is presented. The aim of this review is to highlight the continued interest in this field of study and provide the tools to understand the different unstable mechanisms and techniques used to extend the operating limit.