The general problem of matching structures is very pervasive in computer vision and image processing. The research presented here tackles the problem of object matching in a very general perspective. It is formulated for the matching of surfaces. It applies to objects having small or large deformation and arbitrary topological changes. The process described hinges on a geodesic distance equation for a family of curves or surfaces embedded in the graph of a cost function. This geometrical approach to object matching has the advantage that the similarity criterion can be used to define the shape of the cost function. Matching paths are computed on the cost manifolds using distance maps. These distance maps are generated by solving a general partial differential equation which is a generalization of the geodesic dis- tance evolution scheme introduced by R. Kimmel, A. Amir, and A. F. Bruckstein (1995, IEEE Trans. Pattern Anal. Mach. Intell. 17, 635-640). An Eulerian level-set formulation is also introduced, leading to a numerical scheme used for solving par- tial differential equations originating from hyperbolic conservation laws, which has proven to be very robust and stable.