Toric algebraic geometry provides an extremely successful bridge between the combinatorics of polyhedra and algebraic geometry. Toric varieties provide an experimental testing bed for numerous conjectures in algebraic geometry, and more recently the powerful organizational principles of geometry have found their way into combinatorics, leading to striking breakthroughs. Several classes of varieties have now been defined which promise to generalize the successes of toric varieties. I will describe a class of varieties which correspond to the combinatorics of polyhedra under an additional operation: mutation by piecewise-linear maps. This type of combinatorics appears in work of Escobar and Harada organizing toric degenerations of varieties, as well as work of Rietsch and Williams, and Bossinger, Cheung, Magee, and Nájera Chávez on Newton-Okounkov bodies associated to compactifications of cluster varieties. In these settings, the piecewise linear maps reflect important aspects of the geometry and combinatorics of the associated variety. This is joint work with Escobar, Frias Medina, Harada, and Magee.