During the last few years, a wide variety of Internet of Drones (IoD) applications have emerged with numerous heterogeneous aerial and ground network elements interconnected and equipped with advanced sensors, computation resources, and communication units. The evolution of IoD networks presupposes the mitigation of several security and privacy threats. Thus, robust authentication protocols should be implemented in order to attain secure operation within the IoD. However, owing to the inherent features of the IoD and the limitations of Unmanned Aerial Vehicles (UAVs) in terms of energy, computational, and memory resources, designing efficient and lightweight authentication solutions is a non-trivial and complicated process. Recently, the development of authentication mechanisms for the IoD has received unprecedented attention. In this paper, up-to-date research studies on authentication mechanisms for IoD networks are presented. To this end, the adoption of conventional technologies and methods, such as the widely used hash functions, Public Key Infrastructure (PKI), and Elliptic-Curve Cryptography (ECC), is discussed along with emerging technologies, including Mobile Edge Computing (MEC), Machine Learning (ML), and Blockchain. Additionally, this paper provides a review of effective hardware-based solutions for the identification and authentication of network nodes within the IoD that are based on Trusted Platform Modules (TPMs), Hardware Security Modules (HSMs), and Physically Unclonable Functions (PUFs). Finally, future directions in these relevant research topics are given, stimulating further work.