Amorphous indium gallium zinc oxide (a-IGZO) thin-film transistors (TFTs) have been widely applied in flat-panel displays, flexible electronics, and wearable technologies because of outstanding electrical performance, low power consumption, and low-temperature process compatibility. To bridge the gap between fabrication and circuit-level design, compact models for a-IGZO TFTs have been extensively developed to account for the device characteristics such as current-voltage behavior, capacitance-voltage behavior, and key parameters including field-effect mobility, threshold voltage, and subthreshold swing. These compact models fundamentally rely on the underlying physics of a-IGZO, in particular the nature of density of states (DOS) that play a pivotal role in capturing the physical mechanisms governing device behaviors. DOS modeling enables a deeper understanding of the tail and deep states distributions in the bandgap, which critically influence device stability, reliability, and operational performance. It is also important to explore how process conditions affect DOS profiles and thereby device parameters. This review systematically surveys the progress in physics-based compact modeling for a-IGZO TFTs in the past tw decades since its debut in 2004. Future research directions are also envisioned in developing unified modeling frameworks and advanced simulation techniques to facilitate the development of next-generation a-IGZO-based technologies.