Osmosis is a phenomenon of critical importance in a variety of processes ranging from the transport of ions across cell membranes and the regulation of blood salt levels by the kidneys to the desalination of water and the production of clean energy using potential osmotic power plants. However, despite its importance and over one hundred years of study, there is an ongoing confusion concerning the nature of the microscopic dynamics of the solvent particles in their transfer across the membrane. In this thesis the microscopic dynamical processes underlying osmotic pressure and concentration gradients are investigated using molecular dynamics (MD) simulations. I first present a new derivation for the local pressure that can be used for determining osmotic pressure gradients. Using this result, the steady-state osmotic pressure is studied in a minimal model for an osmotic system and the steady-state density gradients are explained using a simple mechanistic hopping model for the solvent particles. The simulation setup is then modified, allowing us to explore the timescales involved in the relaxation dynamics of the system in the period preceding the steady state. Further consideration is also given to the relative roles of diffusive and non-diffusive solvent transport in this period. Finally, in a novel modi cation to the classic osmosis experiment, the solute particles are driven out-of-equilibrium by the input of energy. The effect of this modi cation on the osmotic pressure and the osmotic ow is studied and we find that active solute particles can cause reverse osmosis to occur. The possibility of defining a new "osmotic effective temperature" is also considered and compared to the results of diffusive and kinetic temperatures.