The importance of radar-based human activity recognition increased significantly over the last two decades in safety and smart surveillance applications due to its superiority towards vision-based sensing in the presence of poor environmental conditions, like illumination, increased radiative heat, occlusion, and fog. An increased public sensitivity for privacy protection, and the progress of cost-effective manufacturing, led to a higher acceptance and distribution. Deep learning approaches proved that the manual feature extraction that relies heavily upon process knowledge can be avoided by its hierarchical, non-descriptive nature. On the other hand, ML techniques based on manual feature extraction provide a robust, yet empirical based approach where the computational effort is comparatively low. This review outlines the basics of classical ML- and DL-based human activity recognition and its advances while taking recent progress of both categories into regard. For every category, state-of-the-art methods are introduced, briefly explained and discussed in terms of their pros, cons and gaps. A comparative study is performed to evaluate the performance and computational effort based on a benchmarking data set to provide a common basis for the assessment of the techniques’ degree of suitability.