Siegel introduced the notion of E-function in a landmark 1929 paper with the goal of generalising the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers. E-functions are power series with algebraic coefficients that are solutions of a linear differential equation and satisfy some growth conditions of arithmetic nature. Besides the exponential function, examples include Bessel functions and a rich family of hypergeometric series. I will explain how such functions arise from geometry in the form of “exponential period functions”, and why it might seem reasonable, in the light of other conjectures, to expect that all E-functions are of this kind.