The propagating shock waves in the solar corona and interplanetary (IP) space caused by fast Coronal Mass Ejections (CMEs) are simulated numerically and their structure and evolution is studied in the framework of ideal magnetohydrodynamics (MHD). Due to the presence of three characteristic velocities and the anisotropy induced by the magnetic field, the CME shocks generated in the lower corona can have a complex structure and topology including secondary shock fronts, over-compressive and compound shocks, etc. The evolution of these CME shocks is followed during their propagation in IP space up to r = 30 R-.. Here, particular attention is given to the effect of the background solar wind on the evolution parameters of the fast CME shocks, i.e. shock speed, deformation of the leading shock front and the CME plasma, stand-off distance of the leading shock front, direction, spread angle, etc. First, different "frequently used" solar wind models are reconstructed with the same numerical code, the same numerical technique on exactly the same numerical grid ( and thus the same numerical dissipation), the same boundary conditions, and the same normalization. Then, a simple CME model is superposed on three different solar wind models, again using exactly the same initial conditions. The result is a fair comparison and thus an objective study of the effect of the background wind on the CME shock evolution. This effect is surprisingly substantial and can be quantified due to the uniformity of the normalization of the used models and simulation techniques.