This paper presents new numerical algorithms, called PSC algorithms, for tracking fronts that propagate with curvature-dependent speed. These algorithms are based on Hamilton-Jacobi formulations and use techniques from hyperbolic conservation laws. The algorithms accurately capture sharp gradients and cusps in moving fronts and naturally handle topological merging and breaking. They work in any number of space dimensions and do not require the moving surface to be expressed as a function. The methods can also be applied to more general Hamilton-Jacobi-type problems. The paper introduces the concept of a Hamilton-Jacobi equation with a right-hand side that depends on curvature effects. This equation is an initial-value Hamilton-Jacobi equation with a right-hand side that depends on curvature effects. The limit of the right-hand side as curvature effects go to zero is an eikonal equation with an associated entropy condition. By viewing the surface as a level set, topological complexities and changes in the moving front are handled naturally. The paper discusses the evolution of the approach, starting with the use of SLIC in a Huyghen's principle flame propagation scheme. It then formulates an entropy condition for moving fronts and shows that the Huyghen's approach is an approximation to the eikonal equation. The paper also discusses the effects of curvature on a propagating front and shows that curvature adds a parabolic right-hand side to the Hamilton-Jacobi equations of motion. The paper presents numerical methods for solving the Hamilton-Jacobi equations, including first-order monotone methods and higher-order methods. These methods are used to approximate the solution to a variety of problems involving propagating curves and surfaces. The paper also discusses the extension of these methods to higher dimensions and the use of non-oscillatory interpolants for high-order accurate approximation. The paper concludes with a discussion of the application of these methods to the problem of front propagation, including the use of level sets and the handling of singularities and far-field boundary conditions. The paper also discusses the addition of passive advection to the front propagation problem and the extension of the algorithms to this important problem.This paper presents new numerical algorithms, called PSC algorithms, for tracking fronts that propagate with curvature-dependent speed. These algorithms are based on Hamilton-Jacobi formulations and use techniques from hyperbolic conservation laws. The algorithms accurately capture sharp gradients and cusps in moving fronts and naturally handle topological merging and breaking. They work in any number of space dimensions and do not require the moving surface to be expressed as a function. The methods can also be applied to more general Hamilton-Jacobi-type problems. The paper introduces the concept of a Hamilton-Jacobi equation with a right-hand side that depends on curvature effects. This equation is an initial-value Hamilton-Jacobi equation with a right-hand side that depends on curvature effects. The limit of the right-hand side as curvature effects go to zero is an eikonal equation with an associated entropy condition. By viewing the surface as a level set, topological complexities and changes in the moving front are handled naturally. The paper discusses the evolution of the approach, starting with the use of SLIC in a Huyghen's principle flame propagation scheme. It then formulates an entropy condition for moving fronts and shows that the Huyghen's approach is an approximation to the eikonal equation. The paper also discusses the effects of curvature on a propagating front and shows that curvature adds a parabolic right-hand side to the Hamilton-Jacobi equations of motion. The paper presents numerical methods for solving the Hamilton-Jacobi equations, including first-order monotone methods and higher-order methods. These methods are used to approximate the solution to a variety of problems involving propagating curves and surfaces. The paper also discusses the extension of these methods to higher dimensions and the use of non-oscillatory interpolants for high-order accurate approximation. The paper concludes with a discussion of the application of these methods to the problem of front propagation, including the use of level sets and the handling of singularities and far-field boundary conditions. The paper also discusses the addition of passive advection to the front propagation problem and the extension of the algorithms to this important problem.