In this thesis we study two different, but related, dynamical systems on hypersurfaces of Euclidean space. First, we consider billiard dynamics in a real-analytic, strictly convex domain. In dimension 2, if a billiard trajectory begins with the velocity vector making a positive angle with the boundary, then this angle remains bounded away from zero at each collision due to the existence of a family of invariant curves called caustics. We show that in higher dimensions, trajectories drifting towards the boundary can exist. Namely, using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, the existence of trajectories asymptotically approaching the billiard boundary is a generic phenomenon in the real- analytic topology. Secondly, we address the assumptions from the first problem. We show that real-analytic, strictly convex, closed surfaces in 3-dimensional Euclidean space generically have a hyperbolic closed geodesic with a transverse homoclinic. It is also shown that real-analytic, closed hypersurfaces with an elliptic closed geodesic generically have a hyperbolic closed geodesic and a transverse homoclinic geodesic, in arbitrary dimension, and without the assumption of strict convexity. These are among the first perturbation-theoretic results for real-analytic geodesic flows.