All viruses that carry a positive-sense RNA genome (+RNA), such as picornaviruses, hepatitis C virus, dengue virus, and SARS- and MERS-coronavirus, confiscate intracellular membranes of the host cell to generate new compartments (i.e., replication organelles) for amplification of their genome. Replication organelles (ROs) are membranous structures that not only harbor viral proteins but also contain a specific array of hijacked host factors that create a unique lipid microenvironment optimal for genome replication. While some lipids may be locally synthesized de novo, other lipids are shuttled towards ROs. In picornavirus-infected cells, lipids are exchanged at membrane contact sites between ROs and other organelles. In this paper, we review recent advances in our understanding of how picornaviruses exploit host membrane contact site machinery to generate ROs, a mechanism that is used by some other +RNA viruses as well.