Uhlmann A. The Metric of Bures and the Geometric Phase. (18K, LaTeX) ABSTRACT. A short synopsis how to define parallel transport along curves of density operators, its minimal length property with respect to the Metric of Bures, and the definition of the associated gauge theory. Some simple examples are given. Slightly extented version of a talk at the First Max Born Symposium, Qroclaw, Polen, May 1991. To appear in the proceedings. Also: Preprint 1-1992 of University of Leipzig, NTZ.