In media with strong spatial dispersion the electric displacement vector and the electric field are typically linked by a partial differential equation in the bulk region. The objective of this work is to highlight that in the vicinity of an interface the relation between the macroscopic fields cannot be univocally determined from the bulk response of the involved materials, but requires instead the knowledge of internal degrees of freedom of the materials. We derive such a relation for the particular case of “wire media”, and describe a numerical formalism that enables characterizing the electromagnetic response of arbitrarily shaped spatially dispersive bodies formed by arrays of crossed wires. The possibility of concentrating the electromagnetic field in a narrow spot by tapering a metamaterial waveguide is discussed