  The aim of this paper is to demonstrate that very many Dehn fillings on a
cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal
and not Seifert fibred, and which has infinite, word hyperbolic fundamental
group. We establish an extension of the Thurston-Gromov $2\pi$ theorem by
showing that if each filling slope has length more than six, then the resulting
3-manifold has all the above properties. We also give a combinatorial version
of the $2\pi$ theorem which relates to angled ideal triangulations. We apply
these techniques by studying surgery along alternating links.
