The random vortex method has been used to simulate the laminar incompressible flow around a circular cylinder oscillating transversely to the flow. The vorticity in the flow is represented by a number of discrete point vortices, which are given random walks to simulate the effect of viscous diffusion. The vortices are created on the cylinder surface to satisfy the no-slip boundary condition, they are convected using an inviscid flow assumption, where the induced velocities on each of the point vortices is obtained using a vortex-in-cell method. Using a time marching algorithim based on the above procedure, the flow around stationary and oscillating circular cylinders is investigated. In particular, the effect of frequency and magnitude of the forced oscillation on the dynamic force coefficients is considered. The results obtained are compared with other numerical solutions and with available experimental data. In general, both the qualitative and quantitative agreement between the present numerical simulation and other numerical results and experiments is good.