This paper is concerned with several not-quantum aspects of black holes, with emphasis on theoretical and mathematical issues related to numerical modeling of black hole space-times. Part of the material has a review character, but some new results or proposals are also presented. We review the experimental evidence for existence of black holes. We propose a definition of black hole region for any theory governed by a symmetric hyperbolic system of equations. Our definition reproduces the usual one for gravity, and leads to the one associated with the Unruh metric in the case of Euler equations. We review the global conditions which have been used in the Scri-based definition of a black hole and point out the deficiencies of the Scri approach. Various results on the structure of horizons and apparent horizons are presented, and a new proof of semi-convexity of horizons based on a variational principle is given. Recent results on the classification of stationary singularity-free vacuum solutions are reviewed. Two new frameworks for discussing black holes are proposed: a "naive approach", based on coordinate systems, and a "quasi-local approach", based on timelike boundaries satisfying a null convexity condition. Some properties of the resulting black holes are established, including an area theorem, topology theorems, and an approximation theorem for the location of the horizon.

43 pages, springer style, several ps files. to appear in the proceedings of the Tuebingen conference on conformal structure of space-time, J. Frauendiener, H. Friedrich, eds, Springer Lecture Notes in Physics