Mobile network is evolving from a communication-only network towards one with joint communication and radar/radio sensing (JCAS) capabilities, that we call perceptive mobile network (PMN). In PMNs, JCAS integrates sensing into communications, sharing a majority of system modules and the same transmitted signals. The PMN is expected to provide a ubiquitous radio sensing platform and enable a vast number of novel smart applications, whilst providing non-compromised communications. In this paper, we present a broad picture of the motivation, methodologies, challenges, and research opportunities of realizing PMN, by providing a comprehensive survey for systems and technologies developed mainly in the last ten years. Beginning by reviewing the work on coexisting communication and radar systems, we highlight their limits on addressing the interference problem, and then introduce the JCAS technology. We then set up JCAS in the mobile network context and envisage its potential applications. We continue to provide a brief review of three types of JCAS systems, with particular attention to their differences in design philosophy. We then introduce a framework of PMN, including the system platform and infrastructure, three types of sensing operations, and signals usable for sensing. Subsequently, we discuss required system modifications to enable sensing on current communication-only infrastructure. Within the context of PMN, we review stimulating research problems and potential solutions, organized under nine topics: performance bounds, waveform optimization, antenna array design, clutter suppression, sensing parameter estimation, resolution of sensing ambiguity, pattern analysis, networked sensing under cellular topology, and sensing-assisted communications. We conclude the paper by listing key open research problems for the aforementioned topics and sharing some lessons that we have learned.