This paper provides an in-depth review of deep learning techniques to address the challenges of odometry and global ego-localization using frequency modulated continuous wave (FMCW) radar sensors. In particular, we focus on the prediction of odometry, which involves the determination of the ego-motion of a system by external sensors, and loop closure detection, which concentrates on the determination of the ego-position typically on an existing map. We initially emphasize the significance of these tasks in the context of radar sensors and underscore the motivations behind them. The subsequent sections delve into the practical implementation of deep learning approaches, strategically designed to effectively address the aforementioned challenges. We primarily focus on spinning and automotive radar configurations within the domain of autonomous driving. Additionally, we introduce publicly available datasets that have been instrumental in addressing these challenges and analyze the importance and struggles of current methods used for radar based odometry and localization. In conclusion, this paper highlights the distinctions between the addressed tasks and other radar perception applications, while also discussing their differences from challenges posed by alternative sensor modalities. The findings contribute to the ongoing discourse on advancing radar sensor capabilities through the application of deep learning methodologies, particularly in the context of enhancing odometry and ego-localization for autonomous driving applications.