Advances in experimental atomic systems have given us access to highly tunable quantum systems, and to an unprecedented range of observables of these systems. One fundamental system that has been made accessible in this way is a gas of bosons trapped in a periodic potential. We present here a several studies of the many-body physics of bosons in optical lattices. We discuss the ferromagnetic effects of a single vacancy in a two-species gas on a lattice. We present a derivation of the superfluid density, the order parameter for the superfluid state, and discuss the mathematical subtleties of calculating it. Finally, we calculate the dynamics of bosonic lattice systems, in the presence of inelastic light scattering used as a density measurement, and during a ramp of the interaction strength from the Mott to the superfluid phase.