Ensembles of ultracold atoms and molecules loaded into lattice potentials are ideally suited to perform model studies of condensed matter systems. In fact, the hope is that ultracold atoms and molecules will some day find use as “quantum simulators” to simulate the properties of a complex many-body system in a fully quantum way.  I will present a series of experiments with ultracold atoms in lattices for we tune the interaction properties. For example, we produce a meta-stable Mott state pinned by attractive interactions. In one-dimensional geometry, we investigate the transition to a pseudo anti-ferromagnetic state. I will outline our attempts to go beyond local interactions by preparing dipolar quantum gases of heteronuclear molecules.