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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2007., Includes bibliographical references (p. 93-95)., Understanding the states in the vicinity of the Mott insulator is crucial to understanding both the physics of the transition between a Mott insulating phase and a metallic phase and the physics of the cuprate high-temperature superconductors. In this thesis, we start from the standard Mott insulating regime of the two dimensional Hubbard model. We then study the physics of nearby states where transport has been restored. First we consider doping of the Hubbard model in the strong coupling limit, i.e. the t-J model. Using the variational Monte Carlo technique, we study Gutzwiller projected states. In particular, studying the projected BCS quasiparticles, we calculate the renormalization of the quasipaticle current and the spectral weight. Both are investigated as a function of momentum and doping. Finally, we discuss the relation between this model and the cuprate superconductors. In the second half of this thesis, we return to the half-filled Hubbard model but now at intermediate values of U/t. In this regime, we study the spin liquid phase, a state that possibly lives between the Mott insulator and the normal metal. Motivated by the recently created organic compound r-(BEDT-TTF)2- Cu2(CN)3, we study a particular spin liquid where there is a spinon Fermi surface coupled to a U(1) gauge field. While still a charge insulator, this model has many metallic-like properties. We first develop a quantum Boltzmann equation for this model from which we calculate the spin resistivity and the more experimentally accessible thermal conductivity. We then proceed to consider spinon pairing and calculate the gauge field contribution to the spin susceptibility. We find that the theoretical result is consistent with experiments giving further evidence that at low temperatures this compound is described by this particular U(1) spin liquid., by Cody Patrick Nave., Ph.D.