||In this thesis, a numerical algorithm based on the integro-differential method is presented to study the unsteady incompressible flow around a circular cylinder. The numerical method can be basically divided into two main parts : kinetics and kinematics. In the kinetics part, the vorticity distribution is examined by solving the vorticity equation using Alternate Direction Implicit technique. The velocity distribution is studied as part of the kinematics procedure. The velocity at the outer boundary is calculated by using an integral relationship which in turn is solved by using Fourier series technique. The velocity in the internal domain is obtained by solving the velocity Poisson equations in the Fourier domain. The surface vorticity is calculated as part of the kinematics step. This is achieved by applying the integral relationship for velocity on the cylinder surface and by imposing the principle of conservation of total vorticity. One of the greatest advantage of the present method is that it can be easily implemented for running on a parallel computer. A parallel version of this is developed on the Intel Paragon computer which is a distributed memory parallel computer. For a problem size of 128 x 128, the best speedup obtainable is 4.67 using 8 processors. After validating against published results for uniform flow around stationary cylinder at Reynolds number Re = 40,300,550 and 1000, the numerical method is applied to study the flow around a rotating cylinder. It shows good agreement with previous result at Re=200, 500 and rotation parameter α at 0.5 and 1.0. For the range of Re between 40 and 200 and for a less than 1, the result shows that the lift is directly proportional to α. The second case investigated is the flow around a circular cylinder in a simple shear flow for Re = 40 to 80 and for shear parameter k less than 0.45. The result indicates that the lift force is acting from the high speed side toward the low speed side of the shear flow which has been controversial for some time. In addition it is found that the mean lift force is proportional to the shear parameter k for the range of Re and k studied.