As taught in basic solid state physics, crystal structure dictates the “band structure” of a material — the energy dispersion relation describing how quantum mechanical electron (Bloch) waves propagate inside it. For many materials, this spectrum, together with its electronic filling as dictated by statistical mechanics, determine the system's equilibrium characteristics such as its electrical and thermal conductivities. Traditionally, band structure has been viewed as a fixed stage upon which the dynamics and responses of a material play out. The advent of new classes of materials and techniques to probe them with external fields has revealed that the electronic properties of materials can be much more malleable than it once seemed. Intriguingly, when such a material is driven out of equilibrium, the electrons within the material itself can produce strong internal fields that can dynamically modify their own underlying band structure. Together we will explore this exciting new world of dynamical quantum matter, in which feedback between the motion of electrons and the underlying fabric that governs their motion leads to striking new phenomena and non-equilibrium states of matter. Following a gentle introduction to the electronic structure of two-dimensional materials, I will illustrate how we can use this new paradigm of non-equilibrium dynamics to unlock novel electromagnetic phenomena that would otherwise be forbidden in any equilibrium system.