Catch bonds are molecular bonds that last longer under force than slip bonds, which become shorter-lived under force. Although catch bonds were initially discovered in studies of leukocyte and bacterial adhesions two decades ago, they have since been found in many other contexts, including platelet binding to blood vessel walls during clotting, structural support within the cell and between cells, force transmission in the cell's machineries for motility and mechanotransduction, viral infection of host cells, and immunoreceptor mechanosensing. Catch bonds are strengthened by increasing force, which induces structural changes in one or both interacting molecules either locally or allosterically to enable additional contacts at their binding interface, thus lengthening bond lifetimes. They can be modeled by the kinetics of a system escaping from the energy well(s) of the bound state(s) over the energy barrier(s) to the free state by traversing along the dissociation path(s) across a hilly energy landscape modulated by force. Catch bond studies are important for understanding the mechanics of biological systems and developing treatment strategies for infectious diseases, immune disorders, cancer, and other ailments.