Caveolae are membrane invaginations with a radius of curvature in the range of 40 nm for the bulb; 10-15 nm is the minimal radius for Lipid bilayers in the liquid-crystalline L-alpha (liquid-disordered: l(d)) phase state. A minimal radius of 20-30 nm could be detected for the gel phase state by analysis of convex-concave bilayer deformations. Circular protrusions with a diameter in the range of only about 40 nm are closed by a flat lid, and those with diameters of 60 nm or more are closed by hemispherical caps. These structures are found primarily in phosphatidylcholine/sterol mixtures, where the gel phase state "liquid ordered" (l(o)) has been introduced. As a further example the mixture of dimyristoylphosphatidylcholine (DMPC) with an unusual sterol (diflucortolon-21-valerat) is presented. In the usual hydration at temperatures above the phase transition the deformation requires an incubation at 4 degrees C for several weeks or months to form. Using a low temperature hydration procedure (at 4 degrees C), surprisingly bilayers of pure DMPC and DPPC (dipalmitoylphosphatidylcholine) are found to deform in the same convex-concave manner, and this takes place within hours and days. The dependence on hydration protocol is also observed for formation of a sponge-like bilayer network with 30-35 nm radius of curvature in brain sphingomyelin and its mixtures with cholesterol. Caveolae are microdomains enriched in cholesterol and sphingomyelin and are simultaneously discussed to be in the l(o) state. Direct evidence by investigation of bilayers formed by the lipids isolated from caveolae is still lacking, but structures similar to caveolae which are in the gel phase state (very probably the l(o) state) are also formed by lipids extracted from bacterial membranes. A further analogy exists because both natural lipid mixtures (brain sphingomyelin and bacterial lipids) transform during heating from the curved bilayer structures into microvesicles above the phase transition. Internalization of caveolae is a process of vesicle formation.