Summary: This paper begins with a short discussion of the general principles of Rigidity Theory. The main interest is the combinatorial part of this subject: generic rigidity. While generic rigidity has several combinatorial characterizations in dimensions one and two, these characterizations have not been able to be extended to characterizations of generic rigidity in higher dimensions. In fact, no ``purely combinatorial'' characterization is presently known for generic rigidity in dimensions three and up. The concept of an abstract rigidity matroid is introduced and, in the context of matroid theory, the present status of the characterization problem is discussed.