a b s t r a c t The application of the AMR–CESE method for solving the Euler and Navier–Stokes equations is presented. The method is a combination of the space–time conservation element and solution element (CESE) method and the adaptive mesh refinement (AMR) technique. Its implementation is based on modification of the original CESE method and utilization of the framework of a parallel-AMR package PARAMESH to manage the block-AMR grid system. Furthermore, a variable time step algorithm is introduced to realize adaptivity of the method in both space and time. A test suite of standard problems for Euler and Navier– Stokes flows are calculated and the results show high resolution, high efficiency and versatility of the method for both shock capture and resolving boundary layer.