Infection of cells by positive-strand RNA viruses generates large numbers of membrane vesicles that provide sites for genome replication. Vesicle formation is initiated by targeting replicase proteins to the cytosolic face of membrane-bound organelles where protein assembly induces membrane curvature. This can result in invagination into the limiting membrane of membrane compartments or induce vesicle budding into the cytoplasm. The new membranes are thought to provide a platform to concentrate proteins, lipids and nucleotides that are required for genome replication. This article describes how recent advances in cell biology and cellular imaging can reveal these structures in 3D, and begin to define how they are formed in terms of effects of specific viral proteins on specific cellular processes.