The fundamental concepts of density-functional theory have been recently applied to minimal-basis model Hamiltonians, in order to describe the physics of strong electron correlation on discrete lattices. The purpose of the present paper is to review current developments on lattice density-functional theory (LDFT) by taking the Anderson model and the Hubbard model as particularly relevant examples for the applications. In LDFT the basic variable is the single-particle density matrix γ _ij with respect to the lattice sites or orbitals i and j. The fundamental unknown functional is the Coulomb-interaction energy W[γ], since the exact expression for the kinetic-energy functional T[γ] is available. Once the general formalism is presented, we derive explicit simple approximations to W[γ] and analyze the performance of LDFT by comparison with exact numerical calculations. Finally, some perspectives of future developments and applications are discussed.