With the development of innovative applications that demand accurate environment information, e.g., autonomous driving, sensing becomes an important requirement for future wireless networks. To this end, integrated sensing and communication (ISAC) provides a promising platform to exploit the synergy between sensing and communication, where perceptive mobile networks (PMNs) were proposed to add accurate sensing capability to existing wireless networks. The well-developed cellular networks offer exciting opportunities for sensing, including large coverage, strong computation and communication power, and most importantly networked sensing, where the perspectives from multiple sensing nodes can be collaboratively utilized for sensing the same target. However, PMNs also face big challenges such as the inherent interference between sensing and communication, the complex sensing environment, and the tracking of high-speed targets by cellular networks. This paper provides a comprehensive review on the design of PMNs, covering the popular network architectures, sensing protocols, standing research problems, and available solutions. Several future research directions that are critical for the development of PMNs are also discussed.