This paper is a survey of some of the developments in coarse extrinsic geometry since
its inception in the work of Gromov. Distortion, as measured by comparing the
diameter of balls relative to different metrics, can be regarded as one of the simplist
extrinsic notions. Results and examples concerning distorted subgroups,
especially in the context of hyperbolic groups and symmetric spaces, are
exposed. Other topics considered are quasiconvexity of subgroups; behaviour
at infinity, or more precisely continuous extensions of embedding maps to
Gromov boundaries in the context of hyperbolic groups acting by isometries on
hyperbolic metric spaces; and distortion as measured using various other filling
invariants.
