"Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. Perlis Many natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc. Copyright © 2006 by the Association for Computing Machinery, Inc.