This paper is concerned with the formulation and development of a numerical moving mesh method to solve time-dependent reaction-diffusion-convection problems. The first part of this contribution gives an over-view of the moving finite element method (MFEM) formulated with a piecewise higher degree polynomial basis in space. In the second part, applications are presented in order to give a convincing demonstration that the proposed moving finite element method is a powerful tool to compute numerical solution of a large class of 1D and 2D problems modeled by time-dependent partial differential equations (PDE). Numerical results are described which illustrate some important features of the proposed moving finite element method for solving problems in one and two-dimensional space domains. (C) 2004 Elsevier Ltd. All rights reserved.