The ongoing amalgamation of UAV and ML techniques is creating a significant synergy and empowering UAVs with unprecedented intelligence and autonomy. This survey aims to provide a timely and comprehensive overview of ML techniques used in UAV operations and communications and identify the potential growth areas and research gaps. We emphasise the four key components of UAV operations and communications to which ML can significantly contribute, namely, perception and feature extraction, feature interpretation and regeneration, trajectory and mission planning, and aerodynamic control and operation. We classify the latest popular ML tools based on their applications to the four components and conduct gap analyses. This survey also takes a step forward by pointing out significant challenges in the upcoming realm of ML-aided automated UAV operations and communications. It is revealed that different ML techniques dominate the applications to the four key modules of UAV operations and communications. While there is an increasing trend of cross-module designs, little effort has been devoted to an end-to-end ML framework, from perception and feature extraction to aerodynamic control and operation. It is also unveiled that the reliability and trust of ML in UAV operations and applications require significant attention before full automation of UAVs and potential cooperation between UAVs and humans come to fruition.