Topologically ordered media demand a new understanding of the emergent properties of quantum matter. This is a fundamental and technological feat. Topological insulators and Weyl semimetals are materials with topological order. Here we will focus on how these materials interact with sources of the electromagnetic field. We start from the effective field theory of Maxwell’s electrodynamics extended by a so-called magnetoelectric term, namely axion electrodynamics and summarize some results we have found exploiting a Green’s function approach to solve for the electromagnetic fields. Signals of the magnetolectric effect are minute compared with other electromagnetic responses, therefore precision measurements are required for its detection. Our formulation can be used for topological insulators and Weyl semimetals with planar, cylindrical and spherical geometries interacting with general charges, currents and boundary conditions. Our formulation is exemplified by: (i) the issue of Casimir effect involving a planar topological insulator, (ii) Vavilov-Cherenkov radiation produced in the forward- and backward-direction of a charged particle traversing a planar interface of two magnetoelectric media and (iii) the electromagnetic fields induced by a static electric charge near the surface of a Weyl semimetal. All three applications can yield observable signals that are within experimental sensitivities.