This talk will provide an overview of some recent work on so-called techniques of "cloaking by mapping". The goal is to surround part of space with a (specially constructed) "cloak" in such a way that the cloak, and any object placed inside the cloak, is invisible (or nearly invisible) to external electromagnetic inspection. I will focus on a particular approximate cloaking scheme, and rigorous estimates for the degree of near-invisibility it provides. The estimates have in order of increasing difficulty been carried out for the Conductivity Problem, the fixed frequency Helmholtz Problem, and the full (time-domain) scalar Wave Problem. I shall try to describe some of the mathematical problems and their solutions in all of these three settings. This work has at various stages been joint with R.V. Kohn, H-M. Nguyen, D. Onofrei and M. Weinstein.