We will show a few applications of the Fourier transform and motivate an extension of this concept, that is literally the Fourier extension operator. This object appears naturally in the study of differential equations that model phenomena in nonlinear optics, plasma physics, biology, fluid mechanics, etc. This will not be a technical talk and our main goal is to understand why certain generic questions in analysis are relevant, such as Does a certain operator map a space X to another space Y? Answering this kind of question is a fundamental step in the study of the phenomena above.