Subanalytic and semialgebraic sets were introduced for topological and systematic investigations of real analytic and algebraic sets. One of the author's purposes is to show that almost all (known and unknown) properties of subanalytic and semialgebraic sets follow abstractly from some fundamental axioms. Another is to develop methods of proof that use finite processes instead of integration of vector fields. The proofs are elementary, but the results obtained are new and significant - for example, for singularity theorists and topologists. Further, the new methods and tools developed provide solid foundations for further research by model theorists (logicians) who are interested in applications of model theory to geometry. A knowledge of basic topology is required.

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.