One-dimensional wires constitute a fundamental building block in nanoscale electronics. However, truly one-dimensional metallic wires do not exist due to Peierls distortion. Molecular wires come close to being stable one-dimensional wires, but are typically semiconductors, with charge transport occurring via tunneling or thermally-activated hopping. In this review, we discuss electron transport through molecular wires, from a theoretical, quantum mechanical perspective based on first principles. We focus specifically on the off-resonant tunneling regime, applicable to shorter molecular wires ({\textless}similar to 4-5 nm) where quantum mechanics dictates electron transport. Here, conductance decays exponentially with the wire length, with an exponential decay constant, beta, that is independent of temperature. Different levels of first principles theory are discussed, starting with the computational workhorse - density functional theory ({DFT}), and moving on to many-electron {GW} methods as well as {GW}-inspired {DFT} + Sigma calculations. These different levels of theory are applied in two major computational frameworks - complex band structure ({CBS}) calculations to estimate the tunneling decay constant, beta, and Landauer-Buttiker transport calculations that consider explicitly the effects of contact geometry, and compute the transmission spectra directly. In general, for the same level of theory, the Landauer-Buttiker calculations give more quantitative values of beta than the {CBS} calculations. However, the {CBS} calculations have a long history and are particularly useful for quick estimates of beta. Comparing different levels of theory, it is clear that {GW} and {DFT} + Sigma calculations give significantly improved agreement with experiment compared to {DFT}, especially for the conductance values. Quantitative agreement can also be obtained for the Seebeck coefficient -another independent probe of electron transport. This excellent agreement provides confirmative evidence of off-resonant tunneling in the systems under investigation. Calculations show that the tunneling decay constant beta is a robust quantity that does not depend on details of the contact geometry, provided that the same contact geometry is used for all molecular lengths considered. However, because conductance is sensitive to contact geometry, values of beta obtained by considering conductance values where the contact geometry is changing with the molecular junction length can be quite different. Experimentally measured values of beta in general compare well with beta obtained using {DFT} + Sigma and {GW} transport calculations, while discrepancies can be attributed to changes in the experimental contact geometries with molecular length. This review also summarizes experimental and theoretical efforts towards finding perfect molecular wires with high conductance and small beta values.

}, keywords = {ab-initio, atomic-force microscopy, Carbon nanotubes, charge-transport, current-voltage characteristics, energy-level alignment, graphene nanoribbons, junction conductance, metal work function, self-assembled monolayers}, issn = {1463-9076}, doi = {10.1039/c4cp05006a}, author = {Khoo, Khoong Hong and Chen, Yifeng and Li, Suchun and Quek, Su Ying} }