A High Altitude Platform Station (HAPS) is a network node that operates in the stratosphere at an altitude around 20 km and is instrumental for providing communication services. Triggered by the technological innovations in the areas of autonomous avionics, array antennas, solar panel efficiency levels and the battery energy density, and fueled by the flourishing industry ecosystems, the HAPS exerts itself as an indispensable component of the next generations of wireless networks. In this article, we provide a vision and framework for the HAPS networks of the future supported by a comprehensive and state-of-the-art literature survey. We highlight the undiscovered potential of HAPS systems, and elaborate on their unique ability to serve metropolitan areas. The latest advancements and promising technologies in the HAPS energy and payload systems are discussed. The integration of the emerging Reconfigurable Smart Surface (RSS) technology in the communications payload of HAPS systems for providing a cost-effective deployment is proposed. A detailed overview of the radio resource management in HAPS systems is presented along with synergistic physical layer techniques, including Faster-Than-Nyquist (FTN) signaling. Numerous aspects of handoff management in HAPS systems are delineated. The notable contribution of Artificial Intelligence (AI) in HAPS, including machine learning in the design, topology management, handoff, and resource allocation aspects are emphasized. The provided extensive overview of the literature is crucial for substantiating our vision that that depicts the expected deployment opportunities and challenges in the next 10 years (next-generation networks), as well as in the subsequent 10 years (next-next-generation networks).