A number of closed-form fundamental solutions for generalized unsteady Oseen and Stokes flows associated with arbitrary time-dependent translational and rotational motions have been developed. These solutions are decomposed into two parts corresponding to a longitudinal wave and a transversal wave. As examples of application, the hydrodynamic forces acting on a sphere and on a circular cylinder translating in an unsteady rotating flow field at low Reynolds numbers are calculated using the generalized fundamental solutions.