This study provides a detailed analysis of current advancements in dynamic object tracking (DOT) and trajectory prediction (TP) methodologies, including their applications and challenges. It covers various approaches, such as feature-based, segmentation-based, estimation-based, and learning-based methods, evaluating their effectiveness, deployment, and limitations in real-world scenarios. The study highlights the significant impact of these technologies in automotive and autonomous vehicles, surveillance and security, healthcare, and industrial automation, contributing to safety and efficiency. Despite the progress, challenges such as improved generalization, computational efficiency, reduced data dependency, and ethical considerations still exist. The study suggests future research directions to address these challenges, emphasizing the importance of multimodal data integration, semantic information fusion, and developing context-aware systems, along with ethical and privacy-preserving frameworks.