This thesis presents a series of theoretical studies of ultra cold atomic systems which model and propose experiments, and develop new computational techniques in order to elucidate aspects of many-body physics and non-equilibrium dynamics. In the first two studies I model the dynamics of nonlinear solitonic excitations in ultracold fermionic superfluids: the first simulates recent experiments and supports the hypothesis that the solitons generated in those experiments are unstable to the formation of vortex rings; the second demonstrates how population imbalance between up and down spin fermions can be used to prevent this instability. In the next study I discuss a method for generating and probing topologically protected edge states using periodically driven optical lattices potentials. Next I use a perturbative approach to study the spectral density of fermions with strong attractive interactions in the normal phase. After that I develop a novel cluster expansion technique to model the dynamics of interacting fermions in a disordered optical lattice. Finally I apply a Ginzurg-Landau theory to model experimental studies of superfluid 3He embedded in nematically ordered aerogel, finding evidence for a new phase of matter --the ``polar phase"-- which is not seen in bulk 3He.