The local matching problem on surfaces is: Given a pair of oriented surfaces in 3-space, find subsurfaces that are identical or complementary in shape. A heuristic method is presented for local matching that is intended for use on complex curved surfaces (rather than such surfaces as as cubes and cylinders). The method proceeds as follows: (1) Find a small set of points-called "critical points" -on the two surfaces with the property that if p is a critical point and p matches q, then q is also a critical point. The critical points are taken to be local extrema of either Gaussian or mean curvature. (2) Construct a rotation invariant representation around each critical point by intersecting the surface with spheres of standard radius centered around the critical point. For each of the resulting curves of intersection, compute a "distance map" function equal to the distance from a point on the curve to the center of gravity of the curve as a. function of arc length (normalized so that the domain of the function is the interval [0,1]). Cll the set of contours for a given critical point a "distance profile." (3) Match distance profiles by computing a "correlation" between corresponding distance contours. (4) Use maximal compatible subsets of the set of matching profiles to induce a transformation that maps corresponding critical points together, then use a cellular spatial partitioning technique to find all points on each surface that are within a tolerance of the other surface.