The transfer of electrons from one entity to another, the former – the electron donor and the latter – the electron acceptor, is one of the most fundamental processes in nature. Important examples being that of electron transfer in photosynthesis,respiration, chemical reactions, photocatalysts, and photovoltaic devices, to name a few. The electron transfer (ET) rate and mechanism is dictated by the strength of the interaction between the donor and acceptor, the temperature, and the external environment. An exact quantum mechanical description is precluded by the exponential scaling of the computation with the number of particles in the system. The path integral formulation to solve the time-dependent Schr¨odinger equation ails from the same problem; however, it is has been used to derive approximate, but rigorous semiclassical theories in real and imaginary time which can incorporate dynamical and statistical quantum mechanical effects such as zero point energy, tunneling, and interference into classical molecular dynamics simulations. For ET occurring at low temperatures, the dominant mechanism is quantum tunneling. The determination of the optimal tunneling pathway – the instanton – and its use in the computation of the ET rate for systems where the electron donor and acceptor are strongly coupled has been extensively studied. Recently, a ring polymer instanton (RPI) – a discrete approximation to the continuous instanton path – was proposed; the difficult trajectory search encountered in semiclassical instanton theory was reformulated as a multidimensional optimization problem. The work presented in this dissertation is geared towards understanding ET at low temperatures, but for systems where the coupling between the electron donor and acceptor is weak, i.e., in the nonadiabatic limit. It elaborates on extending the RPI formulation to multi-state systems, and demonstrates the utility of the multistate ring polymer instanton (MS-RPI) in the computation of nonadiabatic ET rates. Furthermore, the effects of an external bath on the RPI is investigated and its use in the determination of a reaction rate in model systems is demonstrated via both system-bath and reduced dimensional formulations. It is shown that the optimal tunneling path for molecular systems containing conical intersections (accidental degeneracies of the adiabatic eigenstates) can be obtained using the MS-RPI formulation. The discussion of the experimentally observed conductivity of two-dimensional Fe and Cr metal-organic frameworks, and the efforts to explain the them is presented next. This is followed by preliminary results obtained for the extension of the RPI method to multi-dimensional systems, and the use of path sampling methods for RPMD. The thesis concludes with a summary and notes on future directions.