  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
Cohen-Macaulay, 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 PGL(2) and in the two-dimensional building of PGL(3). The latter yields
the classification of Mustafin triangles into 38 combinatorial types.
