Novel air traffic management (ATM) strategies are proposed through the Next Generation Air Transportation and Single European Sky for ATM Research projects to improve the capacity of the airspace and to meet the demands of the future air traffic. The implementation of the proposed solutions leads to increasing use of wireless data for aeronautical communications. Another emerging trend is the unmanned aerial vehicles. The unmanned aerial systems (UASs) need reliable wireless data link and dedicated spectrum allocation for its operation. On-board broadband connectivity also needs dedicated spectrum to satisfy the quality of service requirements of the users. With the growing demand, the aeronautical spectrum is expected to be congested. However, the studies revealed that the aeronautical spectrum is underutilized due to the static spectrum allocation strategy. The aeronautical communication systems, such as air-air and air-ground communication systems, inflight infotainment systems, wireless avionics intra-communications, and UAS, can benefit significantly from the introduction of cognitive radio-based transmission schemes. This paper summarizes the current trends in aeronautical spectrum management followed by the major applications and contributions of cognitive radio in solving the spectrum scarcity crisis in the aeronautical domain. Also, to cope with the evolving technological advancement, researchers have prioritized the issues in the case of cognitive radio that needs to be addressed depending on the domain of operation. The proposed cognitive aeronautical communication systems should also be compliant with the Aeronautical Radio Incorporated and Aerospace Recommended Practice standards. An overview of these standards and the challenges that need immediate attention to make the solution feasible for a large-scale operation, along with the future avenues of research is also furnished.