The boundaries of tracking and sensing solutions are continuously being pushed. A stimulation in this field over recent years is exploiting the properties of millimeter wave (mmWave) radar to achieve simultaneous tracking and sensing of multiple objects. This paper aims to provide a critical analysis of the current literature surrounding multi-object tracking and sensing with short-range mmWave radar. There is significant literature available regarding single-object tracking using mmWave radar, demonstrating the maturity of single-object tracking systems. However, innovative research and advancements are also needed in the field of mmWave radar multi-object tracking, specifically with respect to uniquely identifying multiple target tracks across an interrupted field of view. In this article, we aim to provide an overview of the latest progress in multi-target tracking. In particular, an attempt to phrase the problem space is made by firstly defining a typical multi-object tracking architecture. We then highlight the areas for potential advancements. These areas include sensor fusion, micro-Doppler feature analysis, specialized and generalized activity recognition, gait, tagging and shape profile. Potential multi-object tracking advancements are reviewed and compared with respect to adaptability, performance, accuracy and specificity. Although the majority of the literature reviewed has a focus on human targets, most of the methodologies can be applied to targets consisting of different profiles and characteristics to that of humans. Lastly, future research directions are also discussed to shed light on research opportunities and potential approaches in the open research areas.