Tunneling phospholipid nanotubes between animal cells have recently been identified as a major building block in an important fundamental mechanism of cell-to-cell communication. In order to gain deeper understanding of this interaction and other micro- and nanoscale phenomena connected to material transport and communication in the living world, cell-sized biomimetic devices are required, which need to be structurally or functionally sufficiently close to the living cell. Networks of liposomes and lipid nanotubes are suitable model systems, as they are functionally versatile and structurally highly flexible biomimetic membrane compartments, which allow an effective approach to investigations of chemical synthesis and material transport at the length scale of a biological cell. They are an excellent foundation for detailed studies of cell-to-cell communication, chemical reaction dynamics in confined spaces, macromolecular crowding, exocytosis and other processes critically important for the function of biological cells. In this article we give an overview over the past years of research on nanotube-vesicle networks, introduce briefly fundamental physical and chemical principles, basic and sophisticated experimental techniques of network generation and manipulation, and show application examples and modern approaches to nanofluidic networks that constitute potential future research directions