  We lay the foundations for the study of relatively quasiconvex subgroups of
relatively hyperbolic groups. These foundations require that we first work out
a coherent theory of countable relatively hyperbolic groups (not necessarily
finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's
definitions of relative hyperbolicity for countable groups.
  We then give several equivalent definitions of relatively quasiconvex
subgroups in terms of various natural geometries on a relatively hyperbolic
group. We show that each relatively quasiconvex subgroup is itself relatively
hyperbolic, and that the intersection of two relatively quasiconvex subgroups
is again relatively quasiconvex. In the finitely generated case, we prove that
every undistorted subgroup is relatively quasiconvex, and we compute the
distortion of a finitely generated relatively quasiconvex subgroup.
