A novel adaptive mesh refinement(AMR) strategy based on the Moment-of-Fluid(MOF) method for volume-tracking dynamic interface computation is presented. The Moment-of-Fluid method is a new interface reconstruction and volume advection method using volume fraction as well as material centroid. The adaptive mesh refinement is performed based on the error indicator, the deviation of the actual centroid obtained by interface reconstruction from the reference centroids given by moment advection process. Using the AMR-MOF method, the accuracy of volume-tracking computation with evolving interfaces is improved significantly compared to other published results.