This paper reviews the dynamics of machining and chatter stability research since the first stability laws were introduced by Tlusty and Tobias in the 1950s. The paper aims to introduce the fundamentals of dynamic machining and chatter stability, as well as the state of the art and research challenges, to readers who are new to the area. First, the unified dynamic models of mode coupling and regenerative chatter are introduced. The chatter stability laws in both the frequency and time domains are presented. The dynamic models of intermittent cutting, such as milling, are presented and their stability solutions are derived by considering the time-periodic behavior. The complexities contributed by highly intermittent cutting, which leads to additional stability pockets, and the contribution of the tool's flank face to process damping are explained. The stability of parallel machining operations is explained. The design of variable pitch and serrated cutting tools to suppress chatter is presented. The paper concludes with current challenges in chatter stability of machining which remains to be the main obstacle in increasing the productivity and quality of manufactured parts.