All biological membranes consist of a complex composite of macromolecules and macromolecular assemblies, of which the fluid lipid-bilayer component is a core element with regard to cell encapsulation and barrier properties. The fluid lipid bilayer also supports the functional machinery of receptors, channels and pumps that are associated with the membrane. This bilayer is stabilized by weak physical and colloidal forces, and its nature is that of a self-assembled system of amphiphiles in water. Being only approximately 5 nm in thickness and still encapsulating a cell that is three orders of magnitude larger in diameter, the lipid bilayer as a material has very unusual physical properties, both in terms of structure and dynamics. Although the lipid bilayer is a fluid, it has a distinct and structured trans-bilayer profile, and in the plane of the bilayer the various molecular components, viz different lipid species and membrane proteins, have the capacity to organize laterally in terms of differentiated domains on different length and time scales. These elements of small-scale structure and order are crucial for the functioning of the membrane. It has turned out to be difficult to quantitatively study the small-scale structure of biological membranes. A major part of the insight into membrane micro- and nano-domains and the concepts used to describe them have hence come from studies of simple lipid bilayers as models of membranes, by use of a wide range of theoretical, experimental and simulational approaches. Many questions remain to be answered as to which extent the result from model studies can carry over to real biological membranes.