Power dense electric machines are highly desirable for applications such as electrified transportation systems. One of the factors limiting the achievable power density of electric machines is the thermal management system. In the absence of suitable thermal management systems, the electric machine losses can lead to insulation failure, inter-turn short circuits, de-magnetization of the permanent magnets, and finally drive system failure. Therefore, analyzing the losses in the machine and identifying a suitable thermal management system is essential to realize power dense electric machines. In this paper, thermal analysis techniques applied to electric machines are first reviewed, and their relative merits and demerits are identified. The thermal considerations in electric machine design are discussed. Once the electric machine is thermally analyzed, the next step is to identify or design a suitable thermal management system. The subsequent sections of this paper provide a detailed review of thermal management systems reported in literature for cooling electric machines, with a focus on high power density designs across different application scenarios. Finally, an analysis is presented that demonstrates how the physics of the machine can be used to determine the stator current density limits.