The development of mobile internet and the popularity of mobile terminals produce massive trajectory data of moving objects under the era of big data. Trajectory data has spatio-temporal characteristics and rich information. Trajectory data processing techniques can be used to mine the patterns of human activities and behaviors, the moving patterns of vehicles in the city and the changes of atmospheric environment. However, trajectory data also can be exploited to disclose moving objects' privacy information (e.g., behaviors, hobbies and social relationships). Accordingly, attackers can easily access moving objects' privacy information by digging into their trajectory data such as activities and check-in locations. In another front of research, quantum computation presents an important theoretical direction to mine big data due to its scalable and powerful storage and computing capacity. Applying quantum computing approaches to handle trajectory big data could make some complex problem solvable and achieve higher efficiency. This paper reviews the key technologies of processing trajectory data. First the concept and characteristics of trajectory data is introduced, and the pre-processing methods, including noise filtering and data compression, are summarized. Then, the trajectory indexing and querying techniques, and the current achievements of mining trajectory data, such as pattern mining and trajectory classification, are reviewed. Next, an overview of the basic theories and characteristics of privacy preserving with respect to trajectory data is provided. The supporting techniques of trajectory big data mining, such as processing framework and data visualization, are presented in detail. Some possible ways of applying quantum computation into trajectory data processing, as well as the implementation of some core trajectory mining algorithms by quantum computation are also described. Finally, the challenges of trajectory data processing and promising future research directions are discussed.