Abstract: Transonic flows governed by the Euler equations, about half and full circular cylinders, are investigated numerically. The tool to carry out this study is a finite-difference method based upon an upwind hybrid formulation, a blend of the “lambda” and the “flux-difference splitting” formulations. In the case of the half cylinder, a separation occurs behind the shock, with the generation of a circulating bubble. It is worthwhile mentioning that for M ∞ = 0.5 we have not been able to reach a steady configuration, instead a barely noticeable and perfectly periodic unsteadiness develops. The phenomenon is much more evident at M ∞ = 0.60 . For the full cylinder case, we have found that the symmetrical configuration with respect to the longitudinal geometrical symmetry plane is not stable. Rather, asymmetric, unsteady, periodic flows are predicted with the shedding of eddies behind the cylinder, which trap the vorticity generated by the shocks. Numerical experiments with different computational parameters and grid sizes seem to remove any doubt about the reliability of the present numerical results to represent solutions of the Euler equations in these particular problems.