Expanding Thurston maps were introduced by M. Bonk and D. Meyer with motivation from complex dynamics and Cannon’s conjecture in geometric group theory through Sullivan’s dictionary. In this talk, we explore the ergodic theory of these maps and present recent advances. Specifically, we demonstrate that the entropy map is upper semi-continuous if and only if the map has no periodic critical points. Furthermore, we show that ergodic measures are entropy-dense and derive level-2 large deviation principles for Birkhoff averages, periodic points, and iterated preimages. The proofs employ the thermodynamic formalism for subsystems of expanding Thurston maps, which arise naturally in the study of dynamics on subsets. This work is joint with Zhiqiang Li.