The Fourier transform is introduced, and a pointwise inversion theorem for classical functions is proven. The Fourier transform of tempered distributions is then established. Calculations with Fourier transforms are derived with the corresponding rules regarding symmetries, derivatives, integrals, and convolutions. Typical application examples are generalized Fourier series and impulse sequences, polynomials, and pseudofunctions such as rational functions. Important examples for discrete signal processing are also convolutions of impulse sequences with suitable growth properties of their pulse strengths. The Fourier transforms for square-integrable functions and for functions or distributions with several variables are dealt with in separate sections. Fraunhofer diffraction on rectangular and circular apertures is one of the examples. Further examples on all topics can be found in the text and in the exercises of the chapter.