The use of unmanned aerial vehicles (UAVs) for different applications has increased many folds in recent years. The UAVs are expected to change the future air operations. However, there are instances where the UAVs can be used for malicious purposes. The detection, tracking, and classification of UAVs is challenging compared to manned aerial vehicles (MAVs) mainly due to small size, complex shapes, and ability to fly close to the terrain and in autonomous flight patterns in swarms. In this survey, we will discuss current and future aerial threats, and provide an overview of radar systems to counter such threats. We also study the performance parameters of radar systems for the detection, tracking, and classification of UAVs compared to MAVs. In addition to dedicated radar systems, we review the use of joint communication-radar (JCR) systems, as well as passive monitoring of changes in the common communication signals, e.g., FM, LTE, and any transmissions that may radiate from a UAV, for the detection, tracking, and classification of UAVs are provided. Finally, limitations of radar systems and comparison with other techniques that do not rely on radars for detection, tracking, and classification of aerial threats are provided.