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New electronic devices fabricated from organic molecules have been greatly improved over the past two decades. Yet, understanding the electronic transport mechanism of free carriers and excitons (bound electron-hole pairs) in organic semiconductors (OSs) is still a pertinent challenge. The soft molecular nature of these materials gives rise to an intricate interplay between electronic and nuclear motion as well as unique solid-state physical properties. Standard (analytic) treatments describing electronic transport often rely on one of two extremes: a travelling wave propagating through the material or a particle hopping from one molecular unit to the next. These are often unsuitable to fully describe the complex dynamics, which falls in between these regimes. In this regard, non-adiabatic molecular dynamics simulations permit a direct view into the transport mechanism, thus providing new important insights. In this thesis, I have further developed and improved in terms of efficiency and accuracy a fully atomistic non-adiabatic molecular dynamics algorithm, called fragment orbital-based surface hopping (FOB-SH). This allows the propagation of the coupled electron-nuclear motion in large nano-scale systems. After validating the accuracy of this methodology and discussing important physical requirements (i.e. energy conservation, detailed balance and internal consistency), I will present the application of FOB-SH to the calculation of room temperature charge mobility of a series of molecular organic crystals. I will discuss the agreement with experimental mobility values and the role of the disorder, induced by thermal fluctuations, on the delocalization of the states and the subsequent formation of a polaronic charge state. This polaronic charge propagates through the crystal by diffusive jumps over several lattice spacings at a time during which expands to more than twice its size. I will show that FOB-SH can recover the crossover from hopping to band-like transport depending on the strength of the electronic coupling and the temperature, thus successfully bridging the gap between these two extreme transport regimes. Finally, I will discuss a further extension of FOB-SH to the treatment of exciton transport in OSs. This opens up new exciting avenues for the application of FOB-SH to the study of electronic processes occurring in organic photovoltaic cells.