In the 20th century plastics became an indispensable part of modern life. Most plastic products are produced by melting polymer materials and moulding them into different shapes. The flow or rheological behaviour of molten polymers is highly sensitive to their molecular architectures and molecular weight distributions. Presence of a small amount of long chain branching structures in commercial polymers can alter their rheological and thus processing properties significantly. Therefore a thorough understanding of the relationship between polymer branching and rheology is of crucial importance to the multi-billion pounds plastics industry. The dominant contributions in defining this relationship come from two respects: entanglement effects among long polymer chains or branches and complexity in branching architectures. The entanglement effects originate from the fact that long polymer chains can not pass through each other. As a consequence, the lateral motion of the chains are suppressed, leading to the extremely long relaxation time and characteristic viscoelastic behaviour of entangled polymers, which are qualitatively different from the viscous behaviour of fast relaxing simple liquids. Theoretical works on entanglement dynamics have been for 40 years primarily based on the tube theory. This model assumes that the motion of a linear polymer chain is restricted to a tube-like region along its contour formed by surrounding chains, similar to a snake slithering through an array of obstacles. Recent tube theories can provide appropriate description of the linear rheology of monodisperse linear polymers, but is facing serious difficulties in describing the branched polymers. Synthesized branched polymers can have various architectures, such as star, H-shaped, comb and Cayley-tree polymers. The commercial polymers, such as metallocene polyethylene resins, can even have branches on branches, i.e., hyperbranching, structures. The branching structures prevent these polymers from sliding in the melt as do the linear chains. Instead a star polymer diffuses by retracting its arms all the way to the branch point, allowing this point to move a short distance, and then stretching out the arms again. This is analogous to an octopus entangled in an array of topological constraints (e.g., a fishing net). The relaxation time of stars thus grows exponentially with the length of the arms, in radical contrast to the power law chain-length dependence of the linear polymers. Polymers with more complicated architectures are assumed to relax in a hierarchical way. The relaxation starts from the retraction of the outermost branch arms and proceeds to inner segments layer by layer till the core of the molecule. Theoretical modelling of the branched polymers needs to address several essential questions including the dynamics of the branch arm retraction, the branch point diffusion and the hierarchical relaxation, as well as the reduced entanglement effects caused by the relaxation of surrounding polymers. The fast grow in computer power and simulation techniques enables us to examine these problems in great details. In this project, we propose to perform molecular dynamics simulations to investigate the relaxation dynamics of model branched polymers at the microscopic level. Special attention will be paid to examine and, if needed, re-formulate the assumptions and analytical expressions used in the current tube theories for describing the above-mentioned dynamic processes. Based on these microscopic understanding, more coarse-grained theoretical models will be developed, which will ultimately allow prediction of dynamics and rheology of general mixtures of branched polymers with arbitrary architectures over many decades of time and length scales.