We form networks from aqueous droplets by submerging them in an oil/lipid mixture. When the droplets are joined together, the lipid monolayers surrounding them combine at the interface to form a robust lipid bilayer. Various protein channels and pores can incorporate into the droplet-interface bilayer (DIB), and the application of a potential with electrodes embedded within the droplets allows ionic currents to be driven across the interface and measured. By joining droplets in linear or branched geometries, functional bionetworks can be created. Although the interfaces between neighboring droplets comprise only single lipid bilayers, the structures of the networks are long-lived and robust. Indeed, a single droplet can be "surgically" excised from a network and replaced with a new droplet without rupturing adjacent DIBs. Networks of droplets can be powered with internal "biobatteries" that use ion gradients or the light-driven proton pump bacteriorhodopsin. Besides their interest as coupled protocells, the droplets can be used as devices for ultrastable bilayer recording with greatly reduced electrolyte volume, which will permit their use in rapid screening applications.