The making of an object invisible through some cloaking device is commonly regarded as science fiction. Now two very different cloaking schemes have been proposed. We found that a finite collection of polarizable dipoles within a critical distance of a superlens would become cloaked. This cloaking occurs because the dipoles generate resonant fields which act back on the dipoles and essentially cancel the incident field acting on them. Then Pendry, Schurig and Smith and Leonhardt proposed designing a shield which cloaks objects by guiding electromagnetic waves around them. This generalized an earlier idea of Greenleaf, Lassas and Uhlmann, on cloaking for conductivity. Here we review these developments and also discuss how cloaking might be extended to elasticity using these ideas. This requires new materials, in particular materials with anisotropic density. We show how such materials can be made.