We first provide a quick introduction to the infinite dimensional dynamical systems and then focus on a number of natural questions concerning flows of viscous and incompressible fluids. The main issue is to comprehend the phenomenon of turbulence in fluids and the possibility of description of its several aspects using the model equations of classical hydrodynamics. We are interested in such questions as existence of global in time solutions, existence of global attractors, estimates of their dimensions and the dependence of these estimates on the shape of the domain and the type of boundary data, stability parameters, as well as in comparison of similar hydrodynamical models in these respects. Answers to these questions may serve as tests of soundness and applicability of a given hydrodynamical model and contribute to our understanding of some aspects of turbulent flows.