A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and CohenMacaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PG L(2) and in the 2-dimensional building of PG L(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types. 758 D. Cartwright et al.Fig. 1 Special fibers of Mustafin varieties for d = 3 are degenerations of the projective plane. The two schemes depicted above arise from configurations of n = 4 points in B 3