We propose a mathematical framework to effectively study lattice materials with periodic and non-periodic structures over entire spaces in one, two, and three dimensions. The existence and uniqueness of solutions for periodic lattice problems with absolute terms are proved in discrete Sobolev spaces. By Fourier transform discrete lattice problems are converted to semi-discrete problems for which similar results are established in semi-discrete Sobolev spaces. For lattice problems without absolute terms, additional conditions are imposed on data for the existence and uniqueness of solutions in discrete energy spaces in one, two and three dimensions. Two concrete examples are analyzed in the proposed mathematical framework. The mathematical framework, methodology and techniques in this paper can be utilized or generalized to non-periodic lattices on entire spaces and boundary value problems on lattices.