Lipid-bilayer membranes are formed by self-assembly processes. The molecular interactions within the bilayer and with the environment impart a unique trans-bilayer lateral pressure profile and provide a set of physical mechanisms for formation of lipid domains and laterally differentiated regions in the plane of the membrane. Results from a number of experimental and theoretical studies of model lipid bilayers are reviewed, emphasizing the significance of these fundamental physical properties for the structure and dynamics of biological membranes. Particular attention is paid to the relevance of postulating the existence of equilibrium thermodynamic phases in biological membranes. This includes a discussion of the possible significance of equilibrium critical points in biological membrane systems that normally exist under non-equilibrium conditions. The need for a new model to replace the celebrated Nicolson-Singer fluid-mosaic model of biological membranes is also discussed.