I. Preliminaries. - 1. 1. Whitney stratifications. - 1. 2. Subanalytic sets and semialgebraic sets. - 1. 3. PL topology and C? triangulations. - II. X-Sets. - 11. 1. X-sets. - 11. 2. Triangulations of X-sets. - 11. 3. Triangulations of X functions. - 11. 4. Triangulations of semialgebraic and X0 sets and functions. - 11. 5. Cr X-manifolds. - 11. 6. X-triviality of X-maps. - 11. 7. X-singularity theory. - III. Hauptvermutung For Polyhedra. - III. 1. Certain conditions for two polyhedra to be PL homeomorphic. - III. 2. Proofs of Theorems III. 1. 1 and III. 1. 2. - IV. Triangulations of X-Maps. - IV. l. Conditions for X-maps to be triangulable. - IV. 2. Proofs of Theorems IV. 1. 1 IV. 1. 2 IV. 1. 2? and IV. 1. 2?. - IV. 3. Local and global X-triangulations and uniqueness. - IV. 4. Proofs of Theorems IV. 1. 10 IV. 1. 13 and IV. 1. 13?. - V. D-Sets. - V. 1. Case where any D-set is locally semilinear. - V. 2. Case where there exists a D-set which is not locally semilinear. - List of Notation. Language: English