In his 1989 Lecture Notes, Thurston introduced the notion of self-similar tiling and stated the theorem characterizing expansions for such tilings in the plane as a certain class of algebraic integers (complex Perron numbers). Only the proof of necessity was given. Kenyon, in his 1990 Ph.D. thesis under the direction of Thurston, suggested a generalization to self-affine tilings in higher dimensions; however, his proof was incomplete. I will discuss some progress towards completing this program and open questions that remain.