Externally driven subcritical systems are closely associated with thorium, partially because thorium has no naturally occurring fissile isotopes. Both accelerator-driven systems (ADSs) and fusion-driven systems have been proposed. This paper highlights key literature related to the use of thorium in externally driven systems (EDSs) and builds upon this foundation to identify potential roles for EDSs in thorium fuel cycles. In fuel cycles with natural thorium feed and no enrichment, the potential roles are (1) a once-through breed-and-burn fuel cycle and (2) a fissile breeder (mainly 233U) to support a fleet of critical reactors. If enriched uranium is used in the fuel cycle in addition to thorium, EDSs may be used to burn transuranic material. These fuel cycles were evaluated in the recently completed U.S. Department of Energy Evaluation and Screening of nuclear fuel cycle options relative to the current once-through commercial nuclear fuel cycle in the United States. The evaluation was performed with respect to nine specified high-level criteria, such as waste management and resource utilization. Each of these fuel cycles presents significant potential benefits per unit energy generation compared to the present once-through uranium fuel cycle. A parametric study indicates that fusion-fission-hybrid systems perform better than ADSs in some missions due to a higher neutron source relative to the energy required to produce it. However, both potential externally driven technology choices face significant development and deployment challenges. In addition, there are significant challenges associated with the use of thorium fuel and with the transition from a uranium-based fuel cycle to a thorium-based fuel cycle.