Lattice Density Functional Theory is a powerful method to treat equilibrium structural properties and non-equilibrium kinetics of condensed matter systems. In this thesis an approach based on Markov chains is followed to derive exact density functionals for interacting particles in one-dimension. First, hard rod mixtures on a lattice are considered. For the treatment of this system, certain sets of site occupation numbers are introduced. These sets reflect zero-dimensional or one-particle cavities in continuum treatments, which can hold at most one particle. The exact functional follows from rather simple probabilistic arguments. Thereby the derivation simplifies an earlier, more complicated treatment. A rearrangement of the functional casts it into a form according to lattice fundamental measure theory. This makes it possible to systematically setup approximate density functionals in higher dimensions, which become exact under dimensional reduction. In the next step, the theory is extended to hard rod mixtures with contact interactions. Finally, hard rods with arbitrary nearest-neighbor interactions extending over two rod lengths are studied. For those interactions, two types of zero-dimensional cavities need to be introduced. The first one is a one-particle cavity corresponding to a set of occupation numbers with at most one occupation number being nonzero. The second type is a two-particle cavity, which is a cavity that cannot hold more than two particles, that means at most two occupation numbers can be one in the corresponding set. In order to account for time-dependent kinetics, a lattice version of Time-Dependent Density Functional Theory is followed and applied to hard rods with contact interactions.