Mesoscopic and nano-scale physics describe systems whose dimensions are intermediate between microscopic and macroscopic. Devices manufactured in this size range require the rethinking of electron transport physics. A powerful approach to modeling the nonlinear current-voltage characteristics observed at these scales is the non-equilibrium Green’s functions method of Schwinger, Keldysh, and Craig. In this thesis this approach is used to describe the mesoscopic conductor and spin-dependent transport in ferromagnetic metal-insulator-ferromagnetic metal and ferromagnetic metal-conductor-ferro-magnetic metal junctions. The formalism is also adapted to obtain analytical expressions for the resistance in the integer quantum Hall regime. Also presented is a new formulation of lead-barrier coupling in terms of a self-energy. Financial markets are complex random systems comprised of many interacting investors, each with their own trading strategy and outlook. The task of modeling this complex behavior is akin in complexity to the many body problems in physics. Here non-extensive statistic are used to model price changes in real markets. A risk-neutral valuation model is then derived for derivatives written on such a primary market. This yields a natural generalization of the Black-Scholes derivatives valuation model.