Some recent experiments (see section 1.2) have investigated the effect of gravity on driven granular materials. Gravity is a uniform force field (obviously neglecting the differences of height), therefore it has no effect on the relative velocities, i.e. on the sequence of collisions, at the level of our simple modelization, that is the gas of inelastic hard spheres: it simply accelerates the baricentrum of the gas. Its effect becomes relevant when studied in the presence of particular boundary conditions that break the Galilean invariance, e.g. the walls of the box that contains the material. A wall has a relevant role in the randomization of granular material impinging on it: two particle coming parallel (in a direction not normal to the wall) are reflected in different times and may encounter thereafter; moreover, if the wall is active, i.e. it is vibrating with a random or harmonic movement, the randomization effect is still more efficient and not only internal energy but also total energy is gained. Gravity has eventually the role of driving the grains toward the randomization source (the standing or vibrating walls): in the Du, Li and Kadanoff model (see paragraph 2.5.1) the anomalous or ``extraordinary'' stationary state is observed in the absence of a force field that ensure the efficiency of the thermal wall, the particles are left in a cooling situation (far from the driving wall) for very long times, before the energetic particle come back with its fuel: in a word, almost the whole system is not in contact with the thermal bath for almost all the time, therefore it should not to be surprising to find that the equipartition of energy is violated. When the force of gravity is present, the temperature source is more efficient. However other problems arise, mainly related to the possibility of a hydrodynamic description: we and other authors have shown that the non homogeneity of the gas is essential for the description of the system in the presence of gravity, e.g. it is not true that the temperature is constant all over the system (the ``isothermal atmosphere'' hypothesis); furtherly, fractal clusters, and therefore scale invariance, are present in our measurements, posing doubts on the existence of a mesoscopic scale for hydrodynamic coarse graining. However the equations of hydrodynamics seem to work well in a range of low inelasticity and low density, they are in fair agreement with the density and temperature profiles obtained in DSMC simulations. In this framework, however, it is very difficult to find proper boundary conditions: the energy given to the gas from a vibrating wall depends on the density of the gas near this wall, and the layer near the wall is not correctly described by hydrodynamic equations, as strong spatio-temporal gradients are present. On the other side, at the top free surface of the granular assembly, the grains fly like projectiles, i.e. they rarely collide, and again the hydrodynamics description is useless: to obtain local equilibrium, a high collision rate is essential.