Network reconstruction is a well-developed sub-field of network science, but it has only recently been applied to production networks, where nodes are firms and edges represent customer-supplier relationships. We review the literature that has flourished to infer the topology of these networks by partial, aggregate, or indirect observation of the data. We discuss why this is an important endeavour, what needs to be reconstructed, what makes it different from other network reconstruction problems, and how different researchers have approached the problem. We conclude with a research agenda.