A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier. (C) 2002 Elsevier Science B.V. All rights reserved.