The extensional theology of Newtonian and non-Newtonian fluids is investigated by molecular dynamics simulations of corresponding model liquids in an extending liquid bridge configuration. The fluids are modeled as molecular species and the polymers as chains of monomers bonded by freely-jointed non-linear springs, interacting with atomistic solid walls. We consider three model systems-a Newtonian liquid made of short chains, a polymer solution consisting of longer chains in a Newtonian solvent, and monodisperse and polydisperse polymer melts made of long chains. In the simulations, a cylinder of liquid is placed between solid end-plates, which separate at an exponentially increasing rate until the liquid breaks up. A quantitative description of the time evolution of the liquid filament profile and the forces exerted on the end-plates is obtained, and can be directly compared to experiment. In addition, monitoring of the atomic displacements and forces during the simulation yields information on the internal dynamics of the fluid-the velocity and stress fields, and the molecular configurations. The simulations are in good agreement with laboratory data and with the results of macroscopic numerical calculations where available, but provide new and detailed information on the internal dynamics of liquids in extensional flow.