Some properties of isotropic Brownian and Ornstein Uhlenbeck flows In this thesis we study some macroscopic properties of isotropic Brownian flows (IBF) and isotropic Ornstein Uhlenbeck flows (IOUF) in $ \\R^d $ with $ d\\ge2 $. \\\\ IBFs were studied extensively by Baxendale and Harris and independently by Le Jan. They are a very natural family of stochastic flows with independent and stationary increments and with a distribution which is invariant with respect to spatial rigid motions on $ \\R^d $. First we consider the asymptotics of the evolution of a smooth rectifiable curve under the action of an IBF. We provide almost sure lower and upper exponential bounds for the length of the curve and show that in case the top Lyapunov exponent is strictly negative the lower bound is attained with positive probability. In the case of a positive Lyapunov exponent we do not know if these bounds are attained however we give a sufficient condition concerning the uniformity of the spatial spreading of the curve under which the upper bound is attained a.s.. This might help the further research on the problem. \\\\ Further we consider the evolution of the volume of a set under the action of an IBF. This problem has also been studied by Le Jan Darling and Kunita. Le Jan and Darling proved that in the case of a strictly negative top Lyapunov exponent the volume of a bounded set vanishes almost surely with time (even if its diameter grows). Le Jan has also shown that in the case of a positive Lyapunov exponent the volume does not vanish asymptotically with strictly positive probability. We show that if the IBF is near to the volume preserving case then the volume of an open set does persist (i.e.~does not converge to zero) almost surely. We also study the spatial distribution of the volume of an open set moved by the flow and prove that on a large scale there is a certain kind of asymptotic normality ...