Bruhat-Tits buildings were introduced by Iwahori, Goldman, Matsumoto and formalized by Bruhat-Tits as a tool for studying the structure of p-adic groups appearing in number theory, representation theory, and harmonic analysis. The general theory associates to a semisimple group over a local field (think of SL(n,Q_p)) a (poly-)simplicial complex that encodes much of its structure. In this talk, we shall give an introduction to this circle of ideas and discuss some results surrounding a particular compactification, sometimes called polyhedral or maximal Satake. First introduced by Landvogt, it has been studied under various guises by several authors.