abstract: The recent development of connectivity-altering Monte Carlo algorithms has enabled efficient sampling of the complex configuration spaces of dense, longchain polymers and equilibration of atomistic models of such polymers at all length scales. We will discuss the geometric and statistical mechanical underpinnings of these algorithms, present predictions obtained from them concerning the structure, conformation, and thermodynamic properties of specific polymer melts, and compare these predictions to experimental measurements. We will also discuss the Contour Reduction Topological Analysis (CReTA) algorithm, whereby well-equilibrated atomistic configurations of linear chains are reduced to networks of entangled "primitive paths". The statistical segment length of these primitive paths serves as an excellent estimate of the "tube diameter" invoked in theories of polymer melt deformation and flow.