This paper reviews the state-of-the-art in robotic tails intended for inertial adjustment applications on-board mobile robots. Inspired by biological tails observed in nature, robotic tails provide a separate means to enhance stabilization, and maneuverability from the mobile robot's main form of locomotion, such as legs or wheels. Research over the past decade has primarily focused on implementing single-body rigid pendulum-like tail mechanisms to demonstrate inertial adjustment capabilities on-board walking, jumping and wheeled mobile robots. Recently, there have been increased efforts aimed at leveraging the benefits of both articulated and continuum tail mechanism designs to enhance inertial adjustment capabilities and further emulate the structure and functionalities of tail usage found in nature. This paper discusses relevant research in design, modeling, analysis and implementation of robotic tails onto mobile robots, and highlight how this work is being used to build robotic systems with enhanced performance capabilities. The goal of this article is to outline progress and identify key challenges that lay ahead.