While many aspects of quantum field theory remain a mystery, the mathematics of correlation functions and their renormalization is well understood. My talk will spotlight some of the mathematical aspects which have arisen there. Foremost, it shall be explained how the classic theory of distributions a la Hörmander is applied to problems of quantum field theory, and how renormalization can be understood as a problem of extending distributions. I will then relate how Gabor analysis enters the picture, in particular when one considers so-called twisted products of quantum fields motivated by certain models of noncommutative spaces. The talk requires no knowledge of physics or quantum fields.