The field of ultra-cold atoms, since the achievement of Bose-Einstein Condensation (Anderson et al., 1995; Davis et al., 1995; Bradley et al., 1995), have seen an immensely growing interest over the past decade. With the creation of optical lattices, new possibilities of studying some of the widely used models in condensed matter have opened up. In this dissertation we shall study two such problems, one with two component attractive fermions on optical lattices, and the second one with a circular array of Josephson junctions made with independent BECs. In the first part of the dissertation, we shall study fermions with an attractive interaction in an optical lattice with a single-band Hubbard model away from half-filling with on-site attraction "U" and nearest neighbor hopping "t." Our goal is to understand the crossover from BCS (Bardeen-Cooper-Schrieffer) superfluidity in the weak attraction limit to the BEC of molecules in the strong attraction limit, with particular emphasis on how this crossover in an optical lattice differs from the much better studied continuum problem. We use a large "N" theory with Sp(2"N") symmetry to study the fluctuations beyond mean field theory. At "T" = 0, we calculate across the crossover various observables, including chemical potential, gap, ground state energy, speed of sound and compressibility. The superfluid density "n[subscript s]" is found to have nontrivial "U/t" dependence in this lattice system. We show that the transition temperature "T[subscript c]" scales with the energy gap in the weak coupling limit but crosses over to a "t[superscript 2]/U" scaling in the BEC limit, where phase fluctuations controlled by "n[subscript s]" determine "T[subscript c]". We also find, quite contrary to our expectations, that in the strong coupling limit, the large-"N" theory gives qualitatively wrong trends for compressibility. A comparison with a simple Hartree shifted BCS theory, which takes into account both pairing and Hartree shifts, and correctly recovers the atomic limit and the right qualitative trend for compressibility, reveals that the large-"N" theory on the lattice, although considers a larger number of diagrams, is in fact inferior to the simpler Hartree shifted BCS theory. The failure of the large-"N" approach is explained by noting (i) the importance of Hartree shift in lattice problems, and (ii) inability of the large-"N" approach to treat particle-particle and particle-hole channels at equal footing at the saddle point level. In the second half of the dissertation, we investigate the problem of vortex trapping in cyclically coupled Bose-Josephson junctions. Starting with "N" independent BECs we couple the condensates through Josephson links and allow the system to reach a stable circulation by adding a dissipative term in our semiclassical equations of motion. The central question we address is what is the probability to trap a vortex with winding number "m." Our numerical simulations reveal that the final distribution of winding numbers is narrower than the initial distribution of total phases, indicating an increased probability for no-vortex configurations. Specifically, the final width of the distribution of winding numbers for "N" sites scales as "lambda N alpha," where "alpha" = 0.47 +/- 0.01 and lambda is less than 0.67 (value predicted for the initial distribution). The actual value of "lambda" is found to depend on the strength of dissipation. The nonlinearity of the problem also manifests itself in the result that it is possible to obtain a non-zero circulation starting with zero total phase around the loop. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]