Superconductivity represents one of the most important scientific discoveries of the 20th century. The practical applications are numerous ranging from clean energy storage and MRI machines to quantum computers. However, the low temperatures required for superconductivity prohibits many practical applications. The more recent discovery of high-temperature superconductors, with superconducting transition temperatures above 100~K, has led to the hope that superconductivity at room-temperature might be achievable, although a complete theoretical understanding of the high-temperature superconductors is currently lacking.

In this talk I will review the properties of a specific family of high-temperature superconductors, namely the iron-based materials. While the specific mechanism responsible for the formation of superconducting ground state is unknown, it is believed to be magnetic in nature. Thus, the main focus of my talk will be on the magnetic properties of these materials and how these can be described within specific models.

Applying a simple itinerant approach I will show how the multiorbital Hubbard-Hund Hamiltonian can account for many of the experimentally observed phenomena. I will demonstrate that magnetic fluctuations can drive the system to break rotational symmetry prior to the onset of magnetic order, resulting in so-called nematic order. Furthermore I will discuss how the inclusion of an atomic spin-orbit coupling can explain the observation of a reorientation of the magnetic moments in the tetragonal magnetic phase. Finally, I will consider the case where the magnetic order is incommensurate. This leads to novel magnetic structures and can provide a realisation of intrinsic topological superconductors.