Lipid membranes form the boundary of many biological compartments, including organelles and cells. Consisting of two leaflets of amphipathic molecules, the bilayer membrane forms an impermeable barrier to ions and small molecules. Controlled transport of molecules across lipid membranes is a fundamental biological process that is facilitated by a diverse range of membrane proteins, including ion-channels and pores. However, biological membranes and their associated proteins are challenging to experimentally characterize. These challenges have motivated recent advances in nanotechnology towards building and manipulating synthetic lipid systems. Liposomes—aqueous droplets enclosed by a bilayer membrane—can be synthesised in vitro and used as a synthetic model for the cell membrane. In DNA nanotechnology, DNA is used as programmable building material for self-assembling biocompatible nanostructures. DNA nanostructures can be functionalised with hydrophobic chemical modifications, which bind to or bridge lipid membranes. Here, we review approaches that combine techniques from lipid and DNA nanotechnology to engineer the topography, permeability, and surface interactions of membranes, and to direct the fusion and formation of liposomes. These approaches have been used to study the properties of membrane proteins, to build biosensors, and as a pathway towards assembling synthetic multicellular systems.