Ca/Na montmorillonite and natural Wyoming bentonite (MX-80) have been studied experimentally and theoretically. For a clay system in equilibrium with pure water, Monte Carlo simulations predict a large swelling when the clay counterions are monovalent, while in presence of divalent counterions a limited swelling is obtained with an aqueous layer between the clay platelets of about 10 angstrom. This latter result is in excellent agreement with X-ray scattering data, while dialysis experiments give a significantly larger swelling for Ca montmorillonite in pure water. Obviously, there is one "intra-lamellar" and a second "extra-lamellar" swelling. Montmorillonite in contact with a salt reservoir containing both Na+ and Ca2+ counterfoils will only show a modest swelling unless the Na+ concentration in the hulk is several orders of magnitude larger than the concentration. The limited swelling of clay in presence of divalent counterions is a consequence of correlations, which reduce the entropic repulsion is well as give rise to an attractive component in the total osmotic pressure. Ion ion correlations also favor divalent counterions in a situation with a competition with monovalent ones. A more fundamental result of ion ion correlations is that the osmotic pressure as a function of clay sheet separation becomes nonmonotonic. which indicates the possibility of a phase separation into a concentrated and a dilute clay phase, which would correspond to the "extra-lamellar" swelling found in dialysis experiments. This idea also rinds support in the X-ray scattering spectra, where sometimes two peaks corresponding to different lamellar spacings appear.