This paper i a review on the problem of the motion of incompressible flow around a rotating and translating rigid body D in an exterior domain. D is assumed open and bounded, with Lipschitz boundary. We focus on the asymptotic behavior of the problem. It is shown that the velocity may be split into a constant times the first column of the fundamental solution of the Oseen system, plus a remainder term decaying pointwise near infinity at a rate which is higher than the decay rate of the Oseen tensor. Moreover, it is derived the optimal rates of the velocity, gradient of the velocity and the pressure.