**7. Conclusions**

In this chapter many phenomena have been discussed in order to show the variety of problems which can be found in natural convection of Newtonian and viscoelastic fluids. One of the goals was to show that the different boundary conditions may give results which differ considerably from each other. Sometimes, the results are qualitatively the same and this is taken as an advantage to solve "simpler" problems as those corresponding to the linear and nonlinear equations with free-free boundary conditions. A change in the setting of the problem may produce large complications, as in the case of the free-free boundary conditions, but with one of them being deformable. In this case a new parameter appears, the Galileo number *G*, which complicates not only the number of numerical calculations, but also the physical interpretation of the results, as explained above. As have been shown, the introduction of viscoelasticity complicates even more the physics of convection. Depending on the boundary conditions, there can be stationary and oscillatory cells in linear convection. Nonlinear convection can be stationary but for other magnitudes of the parameters, traveling and standing waves may appear as the stable fluid motion. The problem is to find the conditions and magnitudes of the viscoelastic parameters when a particular convection phenomenon occurs. This is the thrilling part of viscoelastic convection. It is the hope of the present author that this review may motivate a number of readers to work in this rich area of research.
