Trajectory computing is a pivotal domain encompassing trajectory data management and mining, garnering widespread attention due to its crucial role in various practical applications such as location services, urban traffic, and public safety. Traditional methods, focusing on simplistic spatio-temporal features, face challenges of complex calculations, limited scalability, and inadequate adaptability to real-world complexities. In this paper, we present a comprehensive review of the development and recent advances in trajectory computing, from deep learning to the more recent large language models. We first define trajectory data and provide a brief overview of widely-used deep learning models. Systematically, we explore deep learning applications in trajectory management (pre-processing, storage, analysis, and visualization) and mining (trajectory-related forecasting, trajectory-related recommendation, trajectory classification, travel time estimation, anomaly detection, and mobility generation). Furthermore, we discuss emerging research directions and recent advancements in large models (represented by foundation models and large language models) for trajectory computing, which promise to reshape the next generation of trajectory computing. Additionally, we summarize application scenarios, public datasets, and toolkits. Finally, we outline current challenges in trajectory computing research and propose future directions. Relevant papers and open-source resources have been collated and are continuously updated at: https://github.com/yoshall/Awesome-Trajectory-Computing.