In order to compactify proper metric spaces, Gromov introduced the horofunction compactification. For finite-dimensional normed spaces, Walsh explicitely determined the set of Busemann points, which form a subset of the horofunctions. In this talk, we explain the general construction of the horofunction compactification for metric spaces and then focus on finite-dimensional normed spaces with polyhedral norms, for which all horofunctions are Busemann points. We characterize converging sequences in the horofunction compactification and, if time permits, show that the compactification is homeomorphic to the dual unit ball by an explicit map. This result is joint work with Lizhen Ji.