A High Altitude Platform Station (HAPS) is a network node that operates in the stratosphere at an of altitude around 20 km and is instrumental for providing communication services. Precipitated by technological innovations in the areas of autonomous avionics, array antennas, solar panel efficiency levels, and battery energy densities, and fueled by flourishing industry ecosystems, the HAPS has emerged as an indispensable component of 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 review. We highlight the unrealized 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 described. The notable contributions of Artificial Intelligence (AI) in HAPS, including machine learning in the design, topology management, handoff, and resource allocation aspects are emphasized. The extensive overview of the literature we provide is crucial for substantiating our vision 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).