I will start the talk by describing several inverse problems that arise in several contexts. For example, human vision: from the measurements of scattered light that reaches our retinas, our brains construct a detailed three-dimensional map of the world around us. Solving inverse problems is in fact how we obtain a large part of our information about the world we live in. In the second part of the talk, I will discuss invisibility, addressing the question, "can we make objects invisible?" This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc., including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter stories. In the last 14 years or so there have been several scientific efforts to achieve invisibility. I will describe a simple and powerful proposal, the so-called transformation optics and the progress that has been made in achieving invisibility.