This survey examines recent advances in target tracking methods that incorporate neural networks, with a particular emphasis on their application to complex and dynamic tracking scenarios. While classical model-based approaches have traditionally dominated the field, they often struggle with nonlinear dynamics and unpredictable maneuvers. Conversely, learning-based methods, particularly those employing neural architectures, present compelling alternatives by leveraging data-driven representations and adaptive capabilities. This work provides a concise overview of conventional tracking frameworks to contextualize the evolution of neural approaches. A central contribution of the survey is a novel classification of neural tracking methods based on their level of interpretability, offering a unique perspective on how transparency and explainability are addressed in the design of modern tracking systems. The review synthesizes trends across a broad range of applications, compares methodological trade-offs, and identifies key challenges and open research directions, particularly in balancing performance with trustworthiness in real-world deployment.