The concept of magnetic dipolar order -- for example ferro, ferri or antiferromagnetic -- is at the core of our understanding of the behavior of magnetic materials, and is invaluable in explaining their behavior and in selecting and tailoring them for applications. Sometimes, however, magnetic materials surprise us, and behave in ways that can not be captured within our established paradigms, suggesting additional kinds of order that are not yet identified. In this talk I will discuss the relevance of the so-called magnetoelectric multipoles, which form the next-order term, after the magnetic dipole, in the multipolar expansion of the energy of a magnetization density in a magnetic field. First, I will describe how magnetoelectric multipoles underlie multiferroic behavior and in particular how they determine the magnetic response to applied electric fields. Then I will discuss signatures of hidden magnetoelectric multipolar order and possibilities for its direct measurement. I will close with an analysis of how antiferromagnets that contain ordered magnetoelectric multipoles are fundamentally different from those that do not, in particular in their surface magnetic properties.
