  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.
