Inverse problems arise in all fields of science and technology where causes for a desired or observed effect are to be determined. By solving an inverse problem is in fact how we obtain a large part of our information about the world. An example is 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. In the first part of the talk we will describe several inverse problems arising in different contexts.

In the second part of the lecture we will discuss invisibility. 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. In the last 20 years or so there have been several scientific proposals to achieve invisibility. We will describe in a non-technical fashion a simple and powerful proposal, the so-called transformation optics, and the progress that has been made in achieving invisibility.

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