The deployment of 5G infrastructure is one of the main vectors for new application scenarios since it enables enhanced data bandwidth, low latency, and comprehensive signal coverage. As a result, this communication system supports various vertical applications such as the Industrial Internet of Things, smart health, smart cities, smart grids, and transportation systems. However, these applications bring new challenges to 5G networks due to specific requirements for such scenarios. Furthermore, as software-based technologies, including network slicing, software-defined networks, network function virtualization, and multi-access edge computing are a fundamental part of the 5G architecture, the network can expose these applications to new security and privacy concerns. This study summarizes the existing literature on 5G vertical applications security. We highlight vulnerabilities, threats, attacks, and solutions for 5G vertical applications. We conducted a Systematic Literature Mapping (SLM) to discuss security and privacy challenges regarding the 5G vertical applications. The SLM answered the following research question: what is the state-of-the-art regarding security in 5G networks and vertical applications? We reviewed 389 papers from 2,349 produced by searching with a curated search query and discussed vulnerabilities, threats, attacks, and solutions for 5G vertical applications. Smart cities, Industry 4.0, smart transportation, public services, smart grids, and smart health are vertical applications with relevant security concerns. We observed the need for more research since the 5G and vertical applications continuously evolve. This review also uncovers research gaps and future research directions.