Anomaly detection is an active area of research with numerous methods and applications. This survey reviews the state-of-the-art of data-driven anomaly detection techniques and their application to the the aviation domain. After a brief introduction to the main traditional data-driven methods for anomaly detection, we review the recent advances in the area of neural networks, deep learning and temporal-logic based learning. We cover especially unsupervised techniques applicable to time series data because of their relevance to the aviation domain, where the lack of labeled data is the most usual case, and the nature of flight trajectories and sensor data is sequential, or temporal. The advantages and disadvantages of each method are presented in terms of computational efficiency and detection efficacy. The second part of the survey explores the application of anomaly detection techniques to aviation and their contributions to the improvement of the safety and performance of flight operations and aviation systems. As far as we know, some of the presented methods have not yet found an application in the aviation domain. We review applications ranging from the identification of significant operational events in air traffic operations to the prediction of potential aviation system failures for predictive maintenance.