It is argued in this article that the remarkable successes achieved in simulating the flow of Newtonian fluids in complex geometries cannot currently be achieved for most important flows of complex fluids. The examples chosen to illustrate this thesis are the processing of polymer melts, the flow of wormlike micellar surfactant fluids, the flow of suspensions, and the flow of non-Newtonian fluids in porous media. Despite the efforts of rheologists over the past five decades, fully adequate theological equations of state cannot be provided for any of these fluids. This can be attributed to the mesoscale structures (within the fluids viewed as continua) underlying their theological complexity. It is concluded that simulations carried out at several scales, using suitable up-scaling techniques, will be necessary to provide numerical predictions at the process (continuum) level. Empiricism will continue to play a role in process engineering.