The dynamics of entangled flexible polymers is dominated by physics general to many chemical systems. It is an appealing interdisciplinary field where experimental and theoretical physics can work closely with chemistry and chemical engineering. The role of topological interactions is particularly important, and has given rise to a successful theoretical framework: the 'tube model'. Progress over the last 30 years is reviewed in the light of specially-synthesized model materials, an increasing palette of experimental techniques, simulation and both linear and nonlinear rheological response. Our current understanding of a series of processes in entangled dynamics: 'reptation', 'contour length fluctuation' and 'constraint-release' are set in the context of remaining serious challenges. Especial attention is paid to the phenomena associated with polymers of complex topology or 'long chain branching'.