There are a variety of techniques, such as spin images, for comparing the overall shape of one object to another. I am interested in a different problem - how do you determine if this local surface feature is the same as another? This is particularly important in medical applications, where physicians often try to figure out why this shape is different than another. Quantitative comparisons of organic shapes is pretty much an unsolved problem; most techniques rely on length or radii, and that's about it.
Mathematics has used curvature, both Gaussian and Mean, as a way to (very) locally describe shape. Unfortunately, it is both a point-based metric and difficult (if not impossible) to calculate reliably on meshes. Out approach is to look at not just the curvature at a given point, but at curvature in a given area, to quantify shape.
Once we have automatic, reliable methods for determining shape similarity, we can construct 1-1 mappings between shapes that are similar, but not identical (such as the ulna bone from a variety of patients). In this way, we can quantify shape variation across a population.
See also: fitting.