The continuity of abiotically formed bilayer membranes with similar structures in contemporary cellular life, and the requirement for microenvironments in which large and small molecules could be compartmentalized, support the idea that amphiphilic boundary structures contributed to the emergence of life. As an extension of this notion, we propose here a 'Lipid World' scenario as an early evolutionary step in the emergence of cellular life on Earth. This concept combines the potential chemical activities of lipids and other amphiphiles, with their capacity to undergo spontaneous self-organization into supramolecular structures such as micelles and bilayers. In particular, the documented chemical rate enhancements within lipid assemblies suggest that energy-dependent synthetic reactions could lead to the growth and increased abundance of certain amphiphilic assemblies. We further propose that selective processes might act on such assemblies, as suggested by our computer simulations of mutual catalysis among amphiphiles. As demonstrated also by other researchers, such mutual catalysis within random molecular assemblies could have led to a primordial homeostatic system displaying rudimentary life-like properties. Taken together, these concepts provide a theoretical framework, and suggest experimental tests for a Lipid World model for the origin of life.