This paper describes some mathematical models for studying properties of lipid bilayer membranes. It is shown that there is evidence from fluorescent probe and electron spin resonance studies that some integral proteins are "randomly" distributed in the plane of a lipid bilayer for T greater than Tc, so that protein-protein "contacts" can occur. This implies that there is no permanent annulus of lipids around these proteins, so that there is no unique stoichiometric ratio of "boundary lipids" to protein. Computer simulation techniques are reviewed and comments are made about some recently introduced methods. Dynamical models of lipid--integral protein bilayers are outlined and it is shown that much differential scanning calorimetry, freeze-fracture, X-ray, and 2H nuclear magnetic resonance data can be understood. It is predicted that specific heat curves should show a rise at a temperature, TK less than Tc, at which a protein-rich phase starts to "melt"; that dipalmitoyl phosphatidylcholine (DPPC) hydrocarbon chains in such a protein-rich phase should exhibit some static disorder down to approximately -10 degrees C, around which they should freeze noncooperatively; and that the crossing of phase boundaries, as protein concentration expressed as mole fraction X changes just below Tc, should be reflected in a complicated dependence of protein lateral motion upon X. Models to study the liquid condensed-liquid expanded (LC-LE) transition of phosphatidylcholine (PC) monolayers at the air-water interface are described. The rounding in the LC phase of pressure-area isotherms are understood as the melting of microscopic gel-phase "domains" into macroscopic regions of fluid-phase lipids. There is some experimental support for this. Finally, to model the difference between a monolayer and the corresponding bilayer, an interaction is introduced between the two halves of a bilayer. It is shown to be very weak in the case of PC bilayers and it is outlined in the paper how monolayer and bilayer thermodynamics can be related. It was found that the internal lateral pressure of a DPPC bilayer is approximately 30 dyn/cm (1 dyn = 10 microN).