  A group theoretic version of Dehn surgery is studied. Starting with an
arbitrary relatively hyperbolic group $G$ we define a peripheral filling
procedure, which produces quotients of $G$ by imitating the effect of the Dehn
filling of a complete finite volume hyperbolic 3--manifold $M$ on the
fundamental group $\pi_1(M)$. The main result of the paper is an algebraic
counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that
peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and
the group $G$ is approximated (in an algebraic sense) by its quotients obtained
by peripheral fillings. Various applications of these results are discussed.
