We give a complete description of the horofunction boundary of finite-dimensional $\\ell_{p}$ spaces for $1 \\leq p \\leq \\infty$. We also study the variation norm on $\\mathbf{R}^\\mathcal{N}$, $\\mathcal{N}=\\{1,…,N\\}$, and the corresponding horofunction boundary. As a consequence, we describe the horofunctions for Hilbert’s projective metric on the interior of the standard cone $\\mathbf{R}^\\mathcal{N}_{+}$ of $\\mathbf{R}^\\mathcal{N}$.