The theory of Bruhat-Tits buildings has served as a powerful tool in the representation theory of p-adic groups (as, perhaps, the corresponding theory of Bruhat-Tits hovels hopes to play for affine Kac-Moody groups). In this talk, we sketch some of the geometric ideas in buildings used in our proof of the unicity of types for a large class of supercuspidal representations of p-adic groups. This is joint work with Peter Latham, King's College London.