Studies have been underway for several years at General Atomic to assess the capability of the prestressed concrete reactor vessel to contain molten core debris. Analytical techniques have been developed and are being improved. Results of preliminary studies for a 300-MW(e) GCFR demonstration plant indicate that both heat removal downward, through the internal graphite and steel shielding, and upward, primarily by radiation to other shielding, are important. Sensitivity analyses, performed to scope the range of uncertainties in phenomena and materials properties, seem to confirm that critical temperatures are rather insensitive to changes in input parameters. Experiments have been defined that are needed to elucidate and verify thermodynamic processes occurring in the debris, as well as its interactions with shielding and structural materials.