The fifth-generation (5G) New Radio (NR) promises communication services with high reliability, extremely low latency, high capacity, lower complexity, longer battery-life devices, and high user density in order to support the most well-known use cases of latency-aware Ultra Reliable Low Latency Communications (URLLC), unlimited-things-centric Massive Machine Type Communication (mMTC), and bandwidth-devouring enhanced Mobile Broadband (eMBB). To facilitate the exploitation and implementation of this new radio access technology, the so-called private 5G campus networks are expected to become widely used, utilizing multiple access techniques, frequency bands, and the entire underlying wireless infrastructure of public networks for private businesses, vertical industries, and manufacturing. The primary purpose of this type of communication network is to enable businesses, vertical industries, service sectors, universities, and even individuals to take advantage of 5G tailored to their specific activities or to develop their own local networks. Thus, the different advantages of such a technological revolution can be separately exploited by various stakeholders, and at the same time, the scientific community will be able to easily participate in its research and development aimed at addressing its shortcomings. Taking into account both the business and technical benefits, the grand objective of this study is to provide an overview of the security aspects of the private 5G campus networks. To that end, we first focus on the characterization of private 5G campus networks and discuss some background on a number of industrial applications that these types of networks can support. Then, we identify their different security flaws and potential origins. Finally, we highlight several research challenges that need to be addressed.