Three different approaches have been reported in the last few years for identifying topological constraints and generating ensembles of primitive paths (PPs) in entangled, multi-chain polymeric systems. In addition to providing predictions for the static (statistical) properties of the underlying entanglement network, such a development has opened up the way to interfacing atomistic simulation data with reptation, admittedly the most successful phenomenological theory of polymer dynamics for entangled systems. The link between atomistic molecular dynamics simulation results and reptation theory is achieved by geometrically constructing the effective tube around each primitive chain and then documenting chain motion in terms of a curvilinear diffusion inside the effective tube around the coarse-grained chain contour. The outcome of such a topological and dynamical mapping is the computation of the function ψ(s,t), the probability that a segment s of the primitive chain remains inside the initial tube after time t. The purpose of this article is to discuss the new computational developments and to describe how they can be used to advance and improve our current understanding of polymer melt viscoelasticity. We emphasize in particular the opportunity opened today to bring together three different approaches to polymer dynamics (in addition to acquiring reliable experimental data): atomistic simulations, mesoscopic entanglement networks, and tube models. By developing methodologies that can consistently map the results of accurate computer models and simulations of polymer structure and dynamics onto theoretical treatments based on phenomenological concepts (that sometimes defy precise definition), our hope is that we can get a deeper understanding of the predominant relaxation mechanisms in entangled polymers, both in the linear and nonlinear regime, and succeed in our effort to encode them in the form of a constitutive equation for large-scale viscoelastic flow calculations.