5G positioning is an essential part of many applications, such as driverless vehicles, drone tracking, emergency services, location-based services, etc. Such positioning can attain an accuracy on the order of 1 meter or even lower in both indoor and outdoor settings due to its several advantageous aspects, which include millimeter wave signal, multiple input multiple output antennas, device-to-device communication, etc. Hence, many researchers have paid attention to the development of 5G positioning systems over the past decade. This paper, therefore, presents a state-of-the-art review of the existing 5G positioning methods along with their taxonomy and comparative analysis in several metrics including their merits and demerits. Moreover, the architecture and advantageous aspects of 5G positioning are also presented in this paper. Similar to the satellite signals, 5G signals, however, can be impeded by various obstructions, further declining the accuracy of such positioning. Moreover, many security concerns and privacy issues are also associated with 5G positioning. Thus, the development of a precise and secure 5G positioning system faces a number of challenges. Therefore, various ways to improve the 5G positioning accuracy and several solutions to address the security concerns and privacy issues of such positioning are pointed out in this study.