The experimental achievement of Bose-Einstein condensation (1995) and of Fermi degeneracy (1999) in ultra-cold, dilute gases has opened a new field in atomic physics and condensed matter physics. In this thesis, first we present an overview of theoretical and experimental facts on ultra-cold atomic gases. We then describe a Green's function scheme to study coherent transport by fermions through a one-dimensional array of potential wells. Within this scheme different geometries for the array like single-period, double-period and Fibonacci-ordered quasi-periodic array, will be considered. In continue a novel spin-density-functional approach will be introduced to study the ground-state of a one-dimensional trapped Fermi gases inside one-dimensional optical lattices. This approach enables us to investigate both repulsive and attractive Fermi gases within a local-spin-density approximation. We analyze different phases caused by spin-dependent trap for repulsive gas and also by spin-imbalanced population for attractive gas.