The control afforded by Feshbach resonance phenomena has enabled the exploration of strongly interacting degenerate regimes in dilute ultracold atomic alkali-gases. In these dilute systems, interactions are characterized by a single parameter, the s-wave scattering length. In this dissertation, we review the physics of quantum degenerate atomic gases from a theoretical perspective and present the applications of non-perturbative numerical methods ranging from exact diagonalization to quantum Monte Carlo techniques. Emphasis is given to the effect of interactions. A major goal of this work is to compare theoretical predictions with available experimental results.We begin by introducing the effective interactions in the many-body alkali-gas system. As simplifications of the real interaction between two alkali atoms, the resonance properties of various short range models, the zero-range model and the two-channel model are investigated. The fundamental result is that under appropriate conditions, the true interaction potential V(r) of two atoms may be replaced by a regularized delta-function.In collaboration with experimentalists in our department, we developed a unique method to construct localized single-particle wave functions using imaginary time projection and thereby determine lattice Hamiltonian parameters. Our method enables an efficient coarse-grained mapping from a continuum system to a lattice model. We apply the method to a specific disordered potential generated by an optical lattice experiment and calculate for each instance of disorder the equivalent lattice model parameters. Detailed statistical analysis is performed on the resulting probability distributions of the Hubbard parameters.In the final part, we study the pairing and ferromagnetic instabilities in systems of spin-1/2 fermions. To interpret a recent experimental study of the possibility of itinerant ferromagnetism in cold atom systems, we investigate the energy spectrum of a system of four spin-1/2 fermions with short range attractive interactions both exactly and within the scattering length approximation. The formation of molecular bound states and the ferromagnetic transition of the excited scattering state are examined systematically as a function of the two-body scattering length. We show that an adiabatic ferromagnetic transition occurs, but at a critical transition point k_{F}a much higher than predictions from the scattering length approximations. The exact critical interaction strength calculated in the four-particle system is consistent with that reported by experiment. Finally, by constructing a many-body wavefunction satisfying the Bethe-Peierls boundary conditions, we attempted the first variational Monte Carlo calculations of the zero-range model of unitary Fermion gases, eliminating the need for short-range approximations employed by existing QMC calculations.
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