The sound speed profile (SSP) is a core environmental parameter for underwater acoustic detection, navigation, communication, and other applications. However, its accurate acquisition is constrained by the sparsity of observational data and the ill-posed nature of inversion problems. This paper systematically reviews the research progress of SSP inversion under sparse observation constraints: it combs the technical evolution from physical model-driven methods (Matched Field Processing, MFP; Compressed Sensing, CS) to data-driven approaches (Dictionary Learning, DL; Machine Learning, ML), and classifies and compares the principles, applicable scenarios, advantages, and disadvantages of mainstream methods. It integrates typical measured cases from existing studies (including mesoscale eddy monitoring, underwater navigation and positioning, etc.) and quantitatively analyzes the inversion accuracy and practical value of different technical routes. The research shows that fusing physical constraints with multi-source sparse data (remote sensing, in-situ discrete measurements) is the core direction to balance inversion accuracy, efficiency, and cost. This paper provides a comprehensive reference for technical selection in fields such as marine national defense and resource exploration.