Summary: A numerical study is made of unsteady two-dimensional, incompressible flow past an impulsively started translating and rotating circular cylinder. The Reynolds number \(Re\) and the rotating-to-translating speed ratio \(\alpha\) are two controlled parameters, and the influence of their different combinations on vortex shedding from the cylinder is investigated by the numerical scheme sketched below. Associated with the streamfunction-vorticity formulation of Navier-Stokes equations, the Poisson equation for streamfunction is solved by a Fourier/finite-analytic separation-of-variable approach. This approach allows one to attenuate the artificial far-field boundary, and also yields a global conditioning on the wall vorticity in response to the no-slip condition. As for the vorticity transport equation, spatial discretization is done by means of finite difference in which the convection terms are handled with the aid of an ENO (essentially non-oscillatory)-like data reconstruction process. Finally, the interior vorticity is updated by an explicit second-order Runge-Kutta method. Present computations fall into two categories. One with \(Re=10^3\) and \(\alpha\leq 3\); the other with \(Re=10^4\) and \(\alpha\leq 2\). Comparisons with other numerical or physical experiments are included.