Transport of material across the cellular membrane is among the most fundamental biological processes in every living cell. Providing a high-resolution picture of the process is therefore of high relevance to different levels of molecular and cellular biology. Membrane transport can be mediated either by passive diffusion of materials through the lipid phase of the membrane, or facilitated by specialized membrane transport proteins designed for selective and efficient transport of their substrates, either in a passive fashion (e.g., channels and uriporters) or in an active manner (e.g., in active transporters). While novel methodologies in structure determination and biophysical measurements of membrane-associated phenomena have significantly advanced our understanding of some aspects, experimental techniques and approaches do not provide the required high spatial and temporal resolutions to describe a complete atomic-level picture for the underlying dynamics. We have been studying the diverse pathways and mechanisms available for transport of diverse molecular species, and the energetics associated with the phenomena, at the most detailed level (atomic resolution) using an array of complementary computational technologies, including enhanced sampling techniques, free energy calculations, and long-time-scale simulation of realistic biological membranes and membrane proteins. A major challenge on the computational side is to describe the structural changes underlying biological function in lipid bilayers and membrane proteins, which requires sampling high-dimensional free energy landscapes inaccessible to conventional sampling techniques. We have developed novel approaches that, while numerically expensive, have been very efficient to describe structural transitions using non-equilibrium methods employing system-specific collective variables, and a novel combination of several state-of-the-art sampling techniques, using loosely coupled, multiple-copy MD simulations and fully atomistic representations. I will describe the methodology, remaining challenges, and its application to a number of molecular transport phenomena, including passive diffusion of gaseous species (e.g., O2, CO2, NH3), and small molecule nutrients, as well as active transport of material, e.g., proton-coupled sugar transport and ATP medicated drug transport. The results have elucidated highly relevant mechanistic details of membrane transport providing a detailed structural basis for experimentally observed phenomena.