Paul HackingXIE, FEIFEI2024-04-262024-04-262020-022020-0210.7275/15991744https://hdl.handle.net/20.500.14394/18137In this thesis we identify certain cluster varieties with the complement of a union of closures of hypertori in a toric variety. We prove the existence of a compactification $Z$ of the Fock--Goncharov $\mathcal{X}$-cluster variety for a root system $\Phi$ satisfying some conditions, and study the geometric properties of $Z$. We give a relation of the cluster variety to the toric variety for the fan of Weyl chambers and use a modular interpretation of $X(A_n)$ to give another compactification of the $\mathcal{X}$-cluster variety for the root system $A_n$.cluster varietyroot systemtoric varietyAlgebraic Geometrydissertationhttps://orcid.org/0000-0001-6861-4242