The diverse structure and regulated deformation of lipid bilayer membranes are among a cell's most fascinating features. Artificial membrane-bound vesicles, known as liposomes, are versatile tools for modelling biological membranes and delivering foreign objects to cells. To fully mimic the complexity of cell membranes and optimize the efficiency of delivery vesicles, controlling liposome shape (both statically and dynamically) is of utmost importance. Here we report the assembly, arrangement and remodelling of liposomes with designer geometry: all of which are exquisitely controlled by a set of modular, reconfigurable DNA nanocages. Tubular and toroid shapes, among others, are transcribed from DNA cages to liposomes with high fidelity, giving rise to membrane curvatures present in cells yet previously difficult to construct in vitro. Moreover, the conformational changes of DNA cages drive membrane fusion and bending with predictable outcomes, opening up opportunities for the systematic study of membrane mechanics.