These lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume "scissors-congruent", i.e. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristic classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time. Contents: Introduction and History; Scissors Congruence Group and Homology; Homology of Flag Complexes; Translational Scissors Congruences; Euclidean Scissors Congruences; Sydler's Theorem and Non-Commutative Differential Forms; Spherical Scissors Congruences; Hyperbolic Scissors Congruence; Homology of Lie Groups Made Discrete; Invariants; Simplices in Spherical and Hyperbolic 3-Space; Rigidity of Cheeger-Chern-Simons Invariants; Projective Configurations and Homology of the Projective Linear Group; Homology of Indecomposable Configurations; The Case of PGl(3, F). Readership: Graduate students and researchers in geometry and topology.