Heat convection occurs in natural and industrial processes due to the presence of temperature gradients which may appear in any direction with respect to the vertical, which is determined by the direction of gravity. In this case, natural convection is the fluid motion that occurs due to the buoyancy of liquid particles when they have a density difference with respect the surrounding fluid. Here, it is of interest the particular problem of natural convection between two horizontal parallel flat walls. This simple geometry brings about the possibility to understand the fundamental physics of convection. The results obtained from the research of this system may be used as basis to understand others which include, for example, a more complex geometry and a more complex fluid internal structure. Even though it is part of our every day life (it is observed in the atmosphere, in the kitchen, etc.), the theoretical description of natural convection was not done before 1916 when Rayleigh [53] made calculations under the approximation of frictionlesswalls. Jeffreys [27] was the first to calculate the case including friction in the walls. The linear theory can be found in the monograph by Chandrasekhar [7]. It was believed that the patterns (hexagons) observed in the Benard convection (see Fig. 1, in Chapter 2 of [7] and the references at the end of the chapter) were the same as those of natural convection between two horizontal walls. However, it has been shown theoretically and experimentally that the preferred patterns are different. It was shown for the first time theoretically by Pearson [45] that convection may occur in the absence of gravity assuming thermocapillary effects at the free surface of a liquid layer subjected to a perpendicular temperature gradient. The patterns seen in the experiments done by Benard in the year 1900, are in fact only the result of thermocapillarity. The reason why gravity effects were not important is that the thickness of the liquid layer was so small in those experiments that the buoyancy effects can be neglected. As will be shown presently, the Rayleigh number, representative of the buoyancy force in natural convection, depends on the forth power of the thickness of the liquid layer and the Marangoni number, representing thermocapillary effects, depends on the second power of the thickness. This was not realized for more than fifty years, even after the publication of the paper by Pearson (as seen in the monograph by Chandrasekhar). Natural convection may present hexagonal patterns only when non