Buildings are simplicial complexes which serve as combinatorial analogues of many important geometric spaces such as flag manifolds and symmetric spaces. In this talk we shall focus on illustrating many ideas related to buildings via the case of the Bruhat-Tits building associated to $SL(n, F)$, where $F$ is a non-archimedean local field. Such buildings may be viewed as non-archimedean analogues of the (perhaps) more familiar symmetric spaces $SL(n, R)/SO(n)$ (such as the hyperbolic plane). We shall discuss how the group theory of $SL(n, F)$ relates to the geometry of the building, and how the representation theory of $SL(n, F)$ relates to the analysis of functions on the building. Time permitting, we shall also discuss how these ideas show up in recent work of mine regarding "quantum ergodicity for $SL(3, F)$".