With the expansion of wireless mobile networks into both the daily lives of individuals as well as into the widely developing market of connected devices, communication is an increasingly attractive target for attackers. As the complexity of mobile cellular systems grows and the respective countermeasures are implemented to secure data transmissions, the attacks have become increasingly sophisticated on the one hand, but at the same time the system complexity can open up expanded opportunities for security and privacy breaches. After an in-depth summary of possible entry points to attacks to mobile networks, this paper first briefly reviews the basic principles of the physical layer implementation of 4G/5G systems, then gives an overview of possible attacks from a physical layer perspective. It also provides an overview of the software frameworks and hardware tool-software defined radios currently in use for experimenting with 4G/5G mobile networks, and it discusses their basic capabilities. In the final part, the paper summarizes the currently most promising families of techniques to detect illegitimate base stations—the machine-learning-based, localization-based, and behavior-based methods.