This paper deals with the elastodynamic finite integration technique for axisymmetric wave propagation in a homogeneous and heterogeneous cylindrical medium (CEFIT). This special variant of a finite difference time domain (FDTD) scheme offers a suitable method to calculate real three-dimensional problems in a two-dimensional staggered grid. In order to test the accuracy of the numerical CEFIT code, problems for which analytical solutions are available are presented. These solutions involve wave propagation in an elastic plate, the scattering of a plane longitudinal wave by a spherical obstacle, and ultrasound generation by a thermoelastic laser source. For the latter problem experimental results are included. The CEFIT code also allows the treatment of more complicated problems. Further possible applications are the investigation of elastic waves generated in an acoustic microscope, the simulation of impact-echo measurements in multi-layer systems, axisymmetric wave propagation in arbitrary bodies of revolution, the calculation of elastic wave fields of longitudinal wave transducers with a circular aperture, and the investigation of multi-layer models for particulates.