The effectiveness of adaptive space-meshing in the solution of one-dimensional parabolic partial differential equations (PDEs) is assessed. Present day PDE software typically involves discretisation in space (using Finite Differences or Finite Elements) to produce a system of ordinary differential equations (ODEs) which is then solved routinely using currently available high quality ODE integrators. Such approaches do not attempt to control the errors in the spatial discretisation and th e task of ensuring an effective spatial approxim ation and num erical grid are left entirely to the user. Numerical experiments with Burgers’ equation demonstrate the inadequacies of this approach and suggest the need foradaptive spatial m eshing as the problem evolves. The currently used adaptive m eshing techniques for parabolic problems are reviewed and two effective strategies are selected for study. Numerical experim ents dem onstrate their effectiveness in term s of reduced com putational overhead and increased accuracy. From these experiences possible future trends in adaptive meshing can be identified.