The low dimensionality and/or tight confinement realized in an optical lattice can result in a strongly-correlated many-body system. I will describe our investigation of a one dimensional Bose gas which requires a description beyond that which can be provided by mean-field theory. I will also discuss our realization of the 1-D Bose-Hubbard model and our observation of a superfluid-to-Mott insulating transition that exhibits expected as well as unexpected behavior. These and future investigations will help to test the limitations of theoretical techniques developed to treat the many-body problem.