A bead and spring model is considered for the Brownian dynamics simulation of the behavior of cyclic polymer chains (rings) in a dilute solution under shear or elongational flow. Finite extensibility, excluded volume, and hydrodynamic interaction are taken into account to make the polymer model as realistic as possible. In shear flow, the deformation of the chain and the shear rate viscosity dependence (the flow curve), are studied and characterized. In elongational flow, the coil-stretch phenomenon is described and the relationship between the critical elongational rate and the molecular weight is given. The qualitative behavior obtained for ring polymers is analogous to that of linear polymers. (C) 2004 Elsevier Ltd. All rights reserved.