Fracture of materials and interfaces is often time and rate dependent. The underlying mechanisms for the time and rate dependent fracture may include local molecular processes, viscoelasticity, and poroelasticity (solvent diffusion coupled with deformation). In this talk, I will present our recent works on two different mechanisms. First, the effects of poroelasticity on fracture of polymer gels will be discussed. A path-independent, modified J-integral approach is adopted to define the crack-tip energy release rate as the energetic driving force for crack growth in gels, taking into account the energy dissipation by solvent diffusion. For a stationary crack, the energy release rate is time dependent, with which delayed fracture can be predicted based on a Griffith-like fracture criterion. For steady-state crack growth in a long-strip specimen, the energy release rate is a function of the crack speed, with rate-dependent poroelastic toughening. With a poroelastic cohesive zone model, solvent diffusion within the cohesive zone leads to significantly enhanced poroelastic toughening as the crack speed increases. Second, for rate-dependent fracture of a polymer interface, we propose a multiscale cohesive zone model, considering the energetics of bond stretching, the entropic effect of long molecular chains, the kinetics of thermally activated chain scission, and statistical distributions of the chain lengths. This model relates the macroscopically measurable interfacial properties (toughness, strength, and traction-separation relations) to molecular structures of the interface, and the rate dependence results naturally from the kinetics of damage evolution as a thermally activated process. Finite element simulations with the cohesive zone model are directly compared to double cantilever beam experiments for rate-dependent fracture of a silicon/epoxy interface.