Electrification turned over a new leaf in aviation by introducing new types of aerial vehicles along with new means of transportation. Addressing a plethora of use cases, drones are gaining attention in the industry and increasingly appear in the sky. Emerging concepts of flying taxi enable passengers to be transported over several tens of kilometers. Therefore, unmanned traffic management systems are under development to cope with the complexity of future airspace, thereby resulting in unprecedented communication needs. Moreover, the long-term increase in the number of commercial airplanes pushes the limits of voice-oriented communications, and future options such as single-pilot operations demand robust connectivity. In this survey, we provide a comprehensive review and vision for enabling the connectivity applications of aerial vehicles utilizing current and future communication technologies. We begin by categorizing the connectivity use cases per aerial vehicle and analyzing their connectivity requirements. By reviewing more than 500 related studies, we aim for a comprehensive approach to cover wireless communication technologies, and provide an overview of recent findings from the literature toward the possibilities and challenges of employing the wireless communication standards. After analyzing the proposed network architectures, we list the open-source testbed platforms to facilitate future investigations by researchers. This study helped us observe that while numerous works focused on cellular technologies to enable connectivity for aerial platforms, a single wireless technology is not sufficient to meet the stringent connectivity demands of the aerial use cases, especially for the piloting operations. We identified the need of further investigations on multi-technology heterogeneous network architectures to enable robust and real-time connectivity in the sky. Future works should consider suitable technology combinations to develop unified aerial networks that can meet the diverse quality of service demands of the aerial use cases.