New multibeam bathymetry of the entire Puerto Rico trench reveals numerous retrograde slope failures at various scales at the edge of the carbonate platform north of Puerto Rico and the Virgin Islands. The slumped material comprises carbonate blocks, which fail, at least in initial stages, as a coherent rock mass. This, combined with the fact that the edge of the carbonate platform is steeper than most continental slopes, indicates a higher potential for tsunami run-up than along many other U.S. coasts. The style of slope failure appears to be rock falls, slide blocks and debris avalanches. Secondary failure of the failed carbonate products and of the underlying forearc sediments and rocks may lead to debris flows and turbidity flows. Fissures, discovered in the ocean floor near the edge of the platform, indicate that the process is expected to continue in the future. One of the slope failures, the Arecibo amphitheater, previously thought to represent a single giant slide with a volume of 900-1500 cu. km, appears to comprise smaller failures. The expected maximum tsunami run-up on the northern coast of Puerto Rico from one of these slope failures is <20 m, much lower than previously estimated. A recurrence rate of 100s kyr was calculated for the largest observed failures along the edge of the carbonate platform north of Puerto Rico using simple assumptions. Smaller landslides would presumably occur at a higher recurrence rate. Elsewhere around the island, a 20-km wide failure scarp was discovered in the Upper Mona rift and could be associated with the 1918 tsunami and earthquake that hit northwestern Puerto Rico. Large slope failures were also discovered for the first time on the northern side of the Puerto Rico trench. Because Puerto Rico trench slides occur at large water depths (~6000 m), have large horizontal and vertical dimensions, and the directivity from tsunamis emanating from these slides is toward Puerto Rico, they may be of particular concern and may necessitate further study.