Quantum and Classical Simulation of Electronic Transport at the Nanoscale Time-dependent electronic transport is increasingly important to the state-of-the-art device design and fabrication. The development of nanoscale sensing, the harnessing and control of structural fluctuations, and the advancement of next-generation materials all require a treatment of quantum dynamics beyond the level of traditional methods and a more nuanced approach to the quantum/classical divide. It is thus becoming necessary to incorporate new theoretical approaches---as well as efficient computational tools---to fully understand the underlying physical processes in these systems, as well as the approximations used to solve for their behavior. In addition, recent progress in ultra-cold atom experiments allows for the direct observation of many-body transport in the laboratory---a form of quantum simulation---which provides a parallel technique for solving these problems. This thesis focuses on simulation methods for electronic dynamics, from cold-atom to computational approaches. To this end, we examine the use of atomic transport in elucidating the nature of electronic transport and the simulation of the latter in classical computers. In particular, we develop an analog of a scanning tunneling microscope and a corresponding operational meaning of the local density of states for strongly interacting particles---a situation where the concept of quasi-particles cannot often be used. This technique captures the energetic structure of a many-body system through the measurement of particle transport, as well as gives a novel approach to numerically characterize the system. We also demonstrate how interactions can generate steady-state currents in fermionic cold-atom systems, as opposed to globally biased systems. We then shift our attention to the extension of numerical simulations of quantum transport to an open-system formalism---that is, inclusion of an external environment that drives the system out of equilibrium. We include an explicit treatment of the leads of a device with a corresponding finite-time electronic relaxation. This yields a computationally efficient method for simulation of dynamics under non-equilibrium conditions. Moreover, it gives a general simulation technique for finding periodic steady-states, the decay of local disturbances, and the real-time response to structural changes.