The study of twisted two-dimensional (2D) materials, in which twisting layers create moiré superlattices, has created opportunities for investigating topological phases and strongly correlated physics. The broader potential of a seemingly infinite set of twistable 2D materials remains largely unexplored. Here, we define “theoretically twistable materials” as single- or multilayer structures that allow for the construction of simple continuum models of their moiré structures. Our high-throughput algorithm systematically searches for theoretically twistable semimetals and insulators based on the Topological 2D Materials Database (2D-TQCDB). By analyzing key electronic properties, we identify thousands of candidate materials, propose representative twistable materials, and provide examples of crystal growth and exfoliation for several of them. Our results provide a resource for future experimental and theoretical studies of moiré systems.