An implicit, unfactored solver for the simulation of unsteady turbulent flows around moving solid bodies is presented. The method is applied to solve the turbulence transport equations in conjunction with the Navier-Stokes equations in a strong coupled fashion. A third-order accurate upwind scheme is used for discretising the inviscid fluxes and Newton-type sub-iterations are employed to obtain time-accurate solutions on a moving mesh. The flow around an oscillating aerofoil is computed as a test problem and calculations are presented both for laminar and turbulent flows. Initially, laminar flow calculations over an oscillating NACA 0012 aerofoil at various Mach numbers were carried out in order to assess the accuracy of the method in the prediction of the dynamic-stall vortex under laminar flow conditions. Furthermore, calculations were performed for quasi-steady and fully unsteady turbulent flows over the NACA 0015 aerofoil using various turbulence models at different flow conditions corresponding to attached unsteady flow as well as light- and deep-stall cases. It was found that the numerical results for unsteady, turbulent flow calculations over an aerofoil depend strongly on the turbulence model used.