  In 1970, E. M. Andreev published a classification of all three-dimensional
compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a
combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five
classes of linear inequalities, depending on $C$, for the dihedral angles,
which are necessary and sufficient conditions for the existence of a hyperbolic
polyhedron realizing $C$ with the assigned dihedral angles. Andreev's Theorem
also shows that the resulting polyhedron is unique, up to hyperbolic isometry.
  Andreev's Theorem is both an interesting statement about the geometry of
hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof
for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is
also remarkable to what level the proof of Andreev's Theorem resembles (in a
simpler way) the proof of Thurston.
  We correct a fundamental error in Andreev's proof of existence and also
provide a readable new proof of the other parts of the proof of Andreev's
Theorem, because Andreev's paper has the reputation of being ``unreadable''.
