We will start by providing two kinds of motivation for the study of Cannon-Thurston maps: the first classical differential geometric, involving the extrinsic geometry of surfaces in 3-space; the second from complex dynamics involving landing rays. For a hyperbolic subgroup H of a hyperbolic group G, we shall then describe sufficient criteria to guarantee the following two conditions a) Geodesic rays in H starting at 1 land at a unique point of the boundary of G b) The inclusion of H in G does not extend continuously to the boundary As a consequence we obtain diverse classes of examples demonstrating the non-existence of Cannon-Thurston maps. These include the Baker-Riley examples. This is joint work with Rakesh Halder and Pranab Sardar.