Joint communication and radar/radio sensing (JCAS), also known as dual-function radar communications, enables the integration of communication and radio sensing into one system, sharing a single transmitted signal. The perceptive mobile network (PMN) is a natural evolution of JCAS from simple point-to-point links to a mobile/cellular network with integrated radio-sensing capability. In this article, we present a system architecture that unifies three types of sensing, investigate the required modifications to existing mobile networks, and exemplify the signals applicable to sensing. We also provide a review to stimulate research problems and potential solutions, including mutual information, joint design and optimization for waveform and antenna grouping, clutter suppression, sensing parameter estimation and pattern recognition, and networked sensing under the cellular topology.