Since the development of transport systems, humans have exploited ground-level, below-ground, and high-altitude spaces for transportation purposes. However, with the increasing burden of expanding populations and rapid urbanization in recent decades, public transportation systems and freight traffic are suffering huge pressure, plaguing local governments and straining economies. Engineers and researchers have started to re-examine, propose, and develop the underused near-ground spaces (NGS) for transportation purposes. For instance, flying cars, which are not a totally novel idea, aim at solving the traffic congestion problem and releasing the strains on existing city transport networks by utilizing unoccupied NGS. Flying cars differ from traditional grounded transportation systems that are entirely limited by their physical space, such as trains on tracks or automobiles on roads. Flying cars do not occupy or compete for high-altitude spaces used by air traffic for long-distance transfer. As there is limited specific literature on flying cars and flying car transportation systems (FCTS), this paper aims to describe the modern advances, techniques, and challenges of FCTS. We explore the inherent nature of NGS transportation and devise useful proposals to facilitate the construction and commercialization of FCTS. We begin with an introduction to the increasing need for NGS transportation and we address the advantages of using flying cars. Next, we present a brief overview of the history of the development of flying cars in terms of the historic timeline and technique development. Then, we discuss and compare the state of the art in the design of flying cars, including the take-off & landing (TOL) modes, pilot modes, operation modes, and power types, which are related to the adaptability, flexibility & comfort, stability & complexity, and environmental friendliness of flying cars, respectively. Additionally, since large-scale operations of flying cars can improve current transportation problems, we also introduce different facets of the various designs of FCTS, including path and trajectory planning, supporting facilities, and commercial designs. Finally, we discuss the challenges that might arise while developing and commercializing FCTS in terms of safety issues, commercial issues, and ethical issues.