In this talk, I will introduce combinatorial mutations of convex lattice polytopes and discuss their connections with algebraic geometry, symplectic geometry, representation theory, and number theory. I will then focus on mutations of Fano simplices and the invariants they preserve. Finally, I will describe mutations of the Fano simplex associated with P³ and explain their relationship with Q-Gorenstein degenerations of P³. This talk is based in part on joint work with Yongnam Lee.