The present work treats several asymptotic properties of stochastic flows in Euclidean space, whose distributions are frequently assumed to be invariant under rotations of the state space. Despite the fact that the study of some of these flows goes back to the 1950s new models are introduced. The first main chapter treats the asymptotic behavior of the shape of the set of points in the plane that has been visited up to some time by a stochastic flow. It is shown in the case of a planar isotropic Brownian flow that this shape is deterministic in probability. The second main result is the...