The phenomenon of vortex shedding by oscillating cylinders is a complex one. Its understanding is, however, of utmost importance in marine-related engineering, particularly in connection with motions of deep submersibles and marine risers. In this paper, computational results are presented so that the behavior of the shedding as a function of certain parameter space can be elucidated. A methodology based on the random vortex method and a complex-variable boundary-integral formulation is used to study both forced and vortex-induced oscillations of a circular cylinder. Preliminary evaluation of this method indicates that it has been successful in predicting a number of experimentally observed behavior, among which the phenomena of “lock-in” associated with oscillations of the cylinder are well captured.