This paper reviews recent progress in the measurement and modelling of stochastic electromagnetic fields, focusing on propagation approaches based on Wigner functions and the method of moments technique. The respective propagation methods are exemplified by application to measurements of electromagnetic emissions from a stirred, cavity-backed aperture. We discuss early elements of statistical electromagnetics in Heaviside's papers, driven mainly by an analogy of electromagnetic wave propagation with heat transfer. These ideas include concepts of momentum and directionality in the realm of propagation through confined media with irregular boundaries. We then review and extend concepts using Wigner functions to propagate the statistical properties of electromagnetic fields. We discuss in particular how to include polarization in this formalism leading to a Wigner tensor formulation and a relation to an averaged Poynting vector.This article is part of the theme issue 'Celebrating 125 years of Oliver Heaviside's 'Electromagnetic Theory''.