We develop a new surface matching framework to handle surface comparisons based on the mathematical analysis of curves on surfaces, and propose a unique signature for any closed curve on a surface. The signature describes not only the shape of the curve , but also the intrinsic relationship between the curve and its embedding surface; and furthermore, the signature metric is stable across surfaces sharing similar Riemannian geometry metrics. Based on this theoretical advance, we analyze and align features defined as closed curves on surfaces using their signatures. These curves segment a surface into different regions which are mapped onto canonical domains for the matching purpose. The experimental results are very promising, demonstrating that the curve signatures and the comparison framework are robust and discriminative for the effective shape comparison. Besides its utility in our current framework, we believe the curve signature will also serve as a powerful shape segmentation/mapping tool and can be used to aid in many existing techniques towards effective shape analysis.