Modern motivations for extra spacetime dimensions will be presented, in particular the surprising AdS/CFT connection to particle compositeness. It will be shown how highly curved, "warped", extra-dimensional geometries can naturally address several puzzles of fundamental physics, including the weakness of gravity, particle mass hierarchies, dark matter, and supersymmetry breaking. The possibility of direct discovery of warped dimensions at
particle colliders such as the CERN Large Hadron Collider will be discussed. Some current questions in warped cosmology will also be introduced.