Toric varieties are some of the simplest and best-studied objects in algebraic geometry. Many geometric notions have elegant pictorial and combinatorial descriptions in the toric world. I'll describe compactified cluster varieties as a generalization of toric varieties and explain how some of the pictorial and combinatorial gadgets of toric geometry can extend to the cluster setting. In particular, I plan to discuss extensions of following toric geometry topics: fans, polyhedra, convexity, Minkowski sums, and the Batyrev/ Batyrev-Borisov mirror constructions for Calabi-Yau subvarieties of Gorenstein toric Fanos. Some parts of this are worked out while others are still in the naïve guess stage. I'll indicate what we know, what we hope, and what causes complications. Based on collaborations with Lara Bossinger, Mandy Cheung, Juan Bosco Frías Medina, and Alfredo Nájera Chávez.