This report, authored by Stanley Osher and James A. Sethian, presents a new set of numerical algorithms, called PSC (Propagation of Surfaces under Curvature) algorithms, for tracking fronts propagating with curvature-dependent speed. These algorithms approximate the equations of motion, which are similar to Hamilton-Jacobi equations with parabolic right-hand sides, using techniques from hyperbolic conservation laws. The algorithms are designed to handle various curvature-dependent speeds and can be applied to surfaces in any number of space dimensions. They naturally handle topological merging and breaking and do not require the moving surface to be written as a function. The methods are Eulerian and do not suffer from the instability issues associated with marker particle methods or the complexity of volume-of-fluid techniques. The report includes theoretical analysis, numerical methods, and demonstrations of the algorithms through various surface motion problems, such as flame propagation and crystal growth. The algorithms are supported by multiple grants from various organizations, including NASA, NSF, and the Sloan Foundation.This report, authored by Stanley Osher and James A. Sethian, presents a new set of numerical algorithms, called PSC (Propagation of Surfaces under Curvature) algorithms, for tracking fronts propagating with curvature-dependent speed. These algorithms approximate the equations of motion, which are similar to Hamilton-Jacobi equations with parabolic right-hand sides, using techniques from hyperbolic conservation laws. The algorithms are designed to handle various curvature-dependent speeds and can be applied to surfaces in any number of space dimensions. They naturally handle topological merging and breaking and do not require the moving surface to be written as a function. The methods are Eulerian and do not suffer from the instability issues associated with marker particle methods or the complexity of volume-of-fluid techniques. The report includes theoretical analysis, numerical methods, and demonstrations of the algorithms through various surface motion problems, such as flame propagation and crystal growth. The algorithms are supported by multiple grants from various organizations, including NASA, NSF, and the Sloan Foundation.