Symmetric binary fluids, quenched into a regime of immiscibility, undergo phase separation by spinodal decomposition. In the late stages, the fluids are separated by sharply defined, but curved, interfaces: the resulting Laplace pressure drives fluid flow. Scaling ideas (of Siggia and of Furukawa) predict that, ultimately, this flow should become turbulent as inertial effects dominate over viscous ones. The physics here is complex: mesoscale simulation methods (such as lattice Boltzmann and dissipative particle dynamics) can play an essential role in its elucidation, as we describe. Likewise, it is a matter of experience that immiscible fluids will mix, on some lengthscale at least, if stirred vigorously enough. A scaling theory (of Doi and Ohta) predicts the dependence of a steady state domain size on shear rate, but assumes low Reynolds number (inertia is neglected). Our preliminary simulation results (three-dimensional, so far only on small systems) show little sign of the kind of steady state envisaged by Doi and Ohta; they raise instead the possibility of an oriented domain texture which can continue to coarsen until either inertial effects, or (in our simulations) finite size effects, come into play.